let I be non empty set ; :: thesis:
let J be non-Empty Poset-yielding ManySortedSet of I; :: thesis:
set L = product J;
assume A1: for i being Element of I holds
( J . i is up-complete & J . i is lower-bounded ) ; :: thesis:
then reconsider L9 = product J as non empty up-complete Poset by Th11;
let x, y be Element of (product J); :: thesis:

hereby :: thesis:

assume A2: x << y ; :: thesis:

hereby :: thesis:

let i be Element of I; :: thesis:
thus x . i << y . i :: thesis:

proof

let Di be non empty directed Subset of (J . i); :: according to WAYBEL_3:def 1 :: thesis:
assume A3: y . i <= sup Di ; :: thesis:
set di = the Element of Di;
set D = { (y +* (i,z)) where z is Element of (J . i) : z in Di } ;
reconsider di = the Element of Di as Element of (J . i) ;
A4: dom y = I by WAYBEL_3:27;
{ (y +* (i,z)) where z is Element of (J . i) : z in Di } c= the carrier of (product J)

proof

let a be object ; :: according to TARSKI:def 3 :: thesis:
assume a in { (y +* (i,z)) where z is Element of (J . i) : z in Di } ; :: thesis:
then consider z being Element of (J . i) such that
A5: a = y +* (i,z) and
z in Di ;

A6: now :: thesis:

let j be Element of I; :: thesis:
( i = j or i <> j ) ;
then ( ( (y +* (i,z)) . j = z & i = j ) or (y +* (i,z)) . j = y . j ) by A4, FUNCT_7:31, FUNCT_7:32;
hence (y +* (i,z)) . j is Element of (J . j) ; :: thesis:

end;

dom (y +* (i,z)) = I by A4, FUNCT_7:30;
then a is Element of (product J) by A5, A6, WAYBEL_3:27;
hence a in the carrier of (product J) ; :: thesis:

end;

then reconsider D = { (y +* (i,z)) where z is Element of (J . i) : z in Di } as Subset of (product J) ;
A7: y +* (i,di) in D ;
then reconsider D = D as non empty Subset of (product J) ;
D is directed

proof

let z1, z2 be Element of (product J); :: according to WAYBEL_0:def 1 :: thesis:
assume z1 in D ; :: thesis:
then consider a1 being Element of (J . i) such that
A8: z1 = y +* (i,a1) and
A9: a1 in Di ;
assume z2 in D ; :: thesis:
then consider a2 being Element of (J . i) such that
A10: z2 = y +* (i,a2) and
A11: a2 in Di ;
consider a being Element of (J . i) such that
A12: a in Di and
A13: a >= a1 and
A14: a >= a2 by A9, A11, WAYBEL_0:def 1;
y +* (i,a) in D by A12;
then reconsider z = y +* (i,a) as Element of (product J) ;
take z ; :: thesis:
thus z in D by A12; :: thesis:
A15: dom y = I by WAYBEL_3:27;

now :: thesis:

let j be Element of I; :: thesis:
( i = j or i <> j ) ;
then ( ( z . j = a & z1 . j = a1 & i = j ) or ( z . j = y . j & z1 . j = y . j ) ) by A8, A15, FUNCT_7:31, FUNCT_7:32;
hence z . j >= z1 . j by A13, YELLOW_0:def 1; :: thesis:

end;

hence z >= z1 by WAYBEL_3:28; :: thesis:

now :: thesis:

let j be Element of I; :: thesis:
( i = j or i <> j ) ;
then ( ( z . j = a & z2 . j = a2 & i = j ) or ( z . j = y . j & z2 . j = y . j ) ) by A10, A15, FUNCT_7:31, FUNCT_7:32;
hence z . j >= z2 . j by A14, YELLOW_0:def 1; :: thesis:

end;

hence z2 <= z by WAYBEL_3:28; :: thesis:

end;

then reconsider D = D as non empty directed Subset of (product J) ;
A16: dom y = I by WAYBEL_3:27;

now :: thesis:

A17: Di c= pi (D,i)

proof

let a be object ; :: according to TARSKI:def 3 :: thesis:
assume A18: a in Di ; :: thesis:
then reconsider a = a as Element of (J . i) ;
y +* (i,a) in D by A18;
then (y +* (i,a)) . i in pi (D,i) by CARD_3:def 6;
hence a in pi (D,i) by A16, FUNCT_7:31; :: thesis:

end;

let j be Element of I; :: thesis:
A19: ( J . i is non empty up-complete Poset & J . j is non empty up-complete Poset ) by A1;
( not pi (D,i) is empty & pi (D,i) is directed ) by YELLOW16:35;
then A20: ex_sup_of pi (D,i),J . i by A19, WAYBEL_0:75;
ex_sup_of Di,J . i by A19, WAYBEL_0:75;
then A21: sup Di <= sup (pi (D,i)) by A20, A17, YELLOW_0:34;
ex_sup_of D,L9 by WAYBEL_0:75;
then A22: (sup D) . j = sup (pi (D,j)) by YELLOW16:33;

A23: now :: thesis:

assume j <> i ; :: thesis:
then (y +* (i,di)) . j = y . j by FUNCT_7:32;
hence y . j in pi (D,j) by A7, CARD_3:def 6; :: thesis:

end;

( not pi (D,j) is empty & pi (D,j) is directed ) by YELLOW16:35;
then ex_sup_of pi (D,j),J . j by A19, WAYBEL_0:75;
then (sup D) . j is_>=_than pi (D,j) by A22, YELLOW_0:30;
hence (sup D) . j >= y . j by A3, A21, A22, A23, YELLOW_0:def 2; :: thesis:

end;

then sup D >= y by WAYBEL_3:28;
then consider d being Element of (product J) such that
A24: d in D and
A25: d >= x by A2;
consider z being Element of (J . i) such that
A26: d = y +* (i,z) and
A27: z in Di by A24;
take z ; :: thesis:
d . i = z by A4, A26, FUNCT_7:31;
hence ( z in Di & x . i <= z ) by A25, A27, WAYBEL_3:28; :: thesis:

end;

set K = { i where i is Element of I : x . i <> Bottom (J . i) } ;
{ i where i is Element of I : x . i <> Bottom (J . i) } c= I

proof

let a be object ; :: according to TARSKI:def 3 :: thesis:
assume a in { i where i is Element of I : x . i <> Bottom (J . i) } ; :: thesis:
then ex i being Element of I st
( a = i & x . i <> Bottom (J . i) ) ;
hence a in I ; :: thesis:

end;

then reconsider K = { i where i is Element of I : x . i <> Bottom (J . i) } as Subset of I ;
deffunc H₁( Element of I) -> Element of the carrier of (J . $1) = Bottom (J . $1);
consider f being ManySortedSet of I such that
A28: for i being Element of I holds f . i = H₁(i) from PBOOLE:sch 5();

A29: now :: thesis:

let i be Element of I; :: thesis:
f . i = Bottom (J . i) by A28;
hence f . i is Element of (J . i) ; :: thesis:

end;

A30: dom f = I by PARTFUN1:def 2;
then reconsider f = f as Element of (product J) by A29, WAYBEL_3:27;
set X = { (f +* (y | a)) where a is finite Subset of I : verum } ;
{ (f +* (y | a)) where a is finite Subset of I : verum } c= the carrier of (product J)

proof

let a be object ; :: according to TARSKI:def 3 :: thesis:
assume a in { (f +* (y | a)) where a is finite Subset of I : verum } ; :: thesis:
then consider b being finite Subset of I such that
A31: a = f +* (y | b) ;
dom y = I by WAYBEL_3:27;
then A32: dom (y | b) = b by RELAT_1:62;

A33: now :: thesis:

end;

I = I \/ (dom (y | b)) by A32, XBOOLE_1:12
.= dom (f +* (y | b)) by A30, FUNCT_4:def 1 ;
then a is Element of (product J) by A31, A33, WAYBEL_3:27;
hence a in the carrier of (product J) ; :: thesis:

end;

then reconsider X = { (f +* (y | a)) where a is finite Subset of I : verum } as Subset of (product J) ;
f +* (y | ({} I)) in X ;
then reconsider X = X as non empty Subset of (product J) ;
X is directed

proof

let z1, z2 be Element of (product J); :: according to WAYBEL_0:def 1 :: thesis:
assume z1 in X ; :: thesis:
then consider a1 being finite Subset of I such that
A34: z1 = f +* (y | a1) ;
assume z2 in X ; :: thesis:
then consider a2 being finite Subset of I such that
A35: z2 = f +* (y | a2) ;
reconsider a = a1 \/ a2 as finite Subset of I ;
f +* (y | a) in X ;
then reconsider z = f +* (y | a) as Element of (product J) ;
take z ; :: thesis:
thus z in X ; :: thesis:
A36: dom y = I by WAYBEL_3:27;
then A37: dom (y | a) = a by RELAT_1:62;
A38: dom (y | a1) = a1 by A36, RELAT_1:62;

now :: thesis:

let i be Element of I; :: thesis:
A39: f . i = Bottom (J . i) by A28;
J . i is non empty lower-bounded up-complete Poset by A1;
then A40: Bottom (J . i) <= y . i by YELLOW_0:44;

per cases by XBOOLE_0:def 3;

suppose A41: ( not i in a1 & i in a ) ; :: thesis:

then ( z . i = (y | a) . i & (y | a) . i = y . i ) by A37, FUNCT_1:47, FUNCT_4:13;
hence z . i >= z1 . i by A34, A38, A40, A39, A41, FUNCT_4:11; :: thesis:

end;

suppose ( not i in a & not i in a1 ) ; :: thesis:

then ( z . i = f . i & z1 . i = f . i ) by A34, A37, A38, FUNCT_4:11;
hence z . i >= z1 . i by YELLOW_0:def 1; :: thesis:

end;

suppose A42: ( i in a1 & i in a ) ; :: thesis:

then A43: ( z . i = (y | a) . i & (y | a) . i = y . i ) by A37, FUNCT_1:47, FUNCT_4:13;
( z1 . i = (y | a1) . i & (y | a1) . i = y . i ) by A34, A38, A42, FUNCT_1:47, FUNCT_4:13;
hence z . i >= z1 . i by A43, YELLOW_0:def 1; :: thesis:

end;

hence z >= z1 by WAYBEL_3:28; :: thesis:
A44: dom (y | a2) = a2 by A36, RELAT_1:62;

now :: thesis:

let i be Element of I; :: thesis:
A45: f . i = Bottom (J . i) by A28;
J . i is non empty lower-bounded up-complete Poset by A1;
then A46: Bottom (J . i) <= y . i by YELLOW_0:44;

per cases by XBOOLE_0:def 3;

suppose A47: ( not i in a2 & i in a ) ; :: thesis:

then ( z . i = (y | a) . i & (y | a) . i = y . i ) by A37, FUNCT_1:47, FUNCT_4:13;
hence z . i >= z2 . i by A35, A44, A46, A45, A47, FUNCT_4:11; :: thesis:

end;

suppose ( not i in a & not i in a2 ) ; :: thesis:

then ( z . i = f . i & z2 . i = f . i ) by A35, A37, A44, FUNCT_4:11;
hence z . i >= z2 . i by YELLOW_0:def 1; :: thesis:

end;

suppose A48: ( i in a2 & i in a ) ; :: thesis:

then A49: ( z . i = (y | a) . i & (y | a) . i = y . i ) by A37, FUNCT_1:47, FUNCT_4:13;
( z2 . i = (y | a2) . i & (y | a2) . i = y . i ) by A35, A44, A48, FUNCT_1:47, FUNCT_4:13;
hence z . i >= z2 . i by A49, YELLOW_0:def 1; :: thesis:

end;

hence z2 <= z by WAYBEL_3:28; :: thesis:

end;

then reconsider X = X as non empty directed Subset of (product J) ;

now :: thesis:

let i be Element of I; :: thesis:
reconsider a = {i} as finite Subset of I by ZFMISC_1:31;
A50: f +* (y | a) in X ;
then reconsider z = f +* (y | a) as Element of (product J) ;
ex_sup_of X,L9 by WAYBEL_0:75;
then sup X is_>=_than X by YELLOW_0:30;
then A51: z <= sup X by A50;
dom y = I by WAYBEL_3:27;
then A52: dom (y | a) = a by RELAT_1:62;
A53: i in a by TARSKI:def 1;
then (y | a) . i = y . i by FUNCT_1:49;
then z . i = y . i by A53, A52, FUNCT_4:13;
hence (sup X) . i >= y . i by A51, WAYBEL_3:28; :: thesis:

end;

then y <= sup X by WAYBEL_3:28;
then consider d being Element of (product J) such that
A54: d in X and
A55: x <= d by A2;
consider a being finite Subset of I such that
A56: d = f +* (y | a) by A54;
K c= a

proof

dom y = I by WAYBEL_3:27;
then A57: dom (y | a) = a by RELAT_1:62;
let j be object ; :: according to TARSKI:def 3 :: thesis:
assume j in K ; :: thesis:
then consider i being Element of I such that
A58: j = i and
A59: x . i <> Bottom (J . i) ;
J . i is non empty lower-bounded up-complete Poset by A1;
then A60: x . i >= Bottom (J . i) by YELLOW_0:44;
assume not j in a ; :: thesis:
then A61: d . i = f . i by A56, A58, A57, FUNCT_4:11
.= Bottom (J . i) by A28 ;
x . i <= d . i by A55, WAYBEL_3:28;
hence contradiction by A59, A61, A60, ORDERS_2:2; :: thesis:

end;

then reconsider K = K as finite Subset of I ;
take K = K; :: thesis:
thus for i being Element of I st not i in K holds
x . i = Bottom (J . i) ; :: thesis:

end;

defpred S₁[ object , object ] means ex i being Element of I ex z being Element of (product J) st
( $1 = i & $2 = z & z . i >= x . i );
assume A62: for i being Element of I holds x . i << y . i ; :: thesis:
given K being finite Subset of I such that A63: for i being Element of I st not i in K holds
x . i = Bottom (J . i) ; :: thesis:
let D be non empty directed Subset of (product J); :: according to WAYBEL_3:def 1 :: thesis:
assume A64: y <= sup D ; :: thesis:

A65: now :: thesis:

let k be object ; :: thesis:
assume k in K ; :: thesis:
then reconsider i = k as Element of I ;
A66: pi (D,i) is directed

proof

let a, b be Element of (J . i); :: according to WAYBEL_0:def 1 :: thesis:
assume a in pi (D,i) ; :: thesis:
then consider f being Function such that
A67: f in D and
A68: a = f . i by CARD_3:def 6;
assume b in pi (D,i) ; :: thesis:
then consider g being Function such that
A69: g in D and
A70: b = g . i by CARD_3:def 6;
reconsider f = f, g = g as Element of (product J) by A67, A69;
consider h being Element of (product J) such that
A71: ( h in D & h >= f & h >= g ) by A67, A69, WAYBEL_0:def 1;
take h . i ; :: thesis:
thus ( h . i in pi (D,i) & a <= h . i & b <= h . i ) by A68, A70, A71, CARD_3:def 6, WAYBEL_3:28; :: thesis:

end;

ex_sup_of D,L9 by WAYBEL_0:75;
then A72: (sup D) . i = sup (pi (D,i)) by YELLOW16:33;
( x . i << y . i & y . i <= (sup D) . i ) by A62, A64, WAYBEL_3:28;
then consider zi being Element of (J . i) such that
A73: zi in pi (D,i) and
A74: zi >= x . i by A66, A72;
consider a being Function such that
A75: a in D and
A76: zi = a . i by A73, CARD_3:def 6;
reconsider a = a as object ;
take a = a; :: thesis:
thus a in D by A75; :: thesis:
thus S₁[k,a] by A74, A75, A76; :: thesis:

end;

consider F being Function such that
A77: ( dom F = K & rng F c= D ) and
A78: for k being object st k in K holds
S₁[k,F . k] from FUNCT_1:sch 6(A65);
reconsider Y = rng F as finite Subset of D by A77, FINSET_1:8;
consider d being Element of (product J) such that
A79: d in D and
A80: d is_>=_than Y by WAYBEL_0:1;
take d ; :: thesis:
thus d in D by A79; :: thesis:

now :: thesis:

let i be Element of I; :: thesis:
A81: J . i is non empty lower-bounded up-complete Poset by A1;

per cases ;

suppose A82: i in K ; :: thesis:

then consider j being Element of I, z being Element of (product J) such that
A83: i = j and
A84: F . i = z and
A85: z . j >= x . j by A78;
z in Y by A77, A82, A84, FUNCT_1:def 3;
then d >= z by A80;
then d . i >= z . i by WAYBEL_3:28;
hence d . i >= x . i by A81, A83, A85, YELLOW_0:def 2; :: thesis:

end;

suppose not i in K ; :: thesis:

then x . i = Bottom (J . i) by A63;
hence d . i >= x . i by A81, YELLOW_0:44; :: thesis:

end;

hence x <= d by WAYBEL_3:28; :: thesis: