let S be non empty non void ManySortedSign ; for U0 being MSAlgebra over S
for U1, U2 being MSSubAlgebra of U0 st the Sorts of U1 c= the Sorts of U2 holds
U1 is MSSubAlgebra of U2
let U0 be MSAlgebra over S; for U1, U2 being MSSubAlgebra of U0 st the Sorts of U1 c= the Sorts of U2 holds
U1 is MSSubAlgebra of U2
let U1, U2 be MSSubAlgebra of U0; ( the Sorts of U1 c= the Sorts of U2 implies U1 is MSSubAlgebra of U2 )
reconsider B1 = the Sorts of U1, B2 = the Sorts of U2 as MSSubset of U0 by Def9;
assume A1:
the Sorts of U1 c= the Sorts of U2
; U1 is MSSubAlgebra of U2
hence
the Sorts of U1 is MSSubset of U2
by PBOOLE:def 18; MSUALG_2:def 9 for B being MSSubset of U2 st B = the Sorts of U1 holds
( B is opers_closed & the Charact of U1 = Opers (U2,B) )
let B be MSSubset of U2; ( B = the Sorts of U1 implies ( B is opers_closed & the Charact of U1 = Opers (U2,B) ) )
A2:
B1 is opers_closed
by Def9;
set O = the Charact of U1;
set P = Opers (U2,B);
A3:
the Charact of U1 = Opers (U0,B1)
by Def9;
A4:
B2 is opers_closed
by Def9;
A5:
the Charact of U2 = Opers (U0,B2)
by Def9;
assume A6:
B = the Sorts of U1
; ( B is opers_closed & the Charact of U1 = Opers (U2,B) )
A7:
for o being OperSymbol of S holds B is_closed_on o
proof
let o be
OperSymbol of
S;
B is_closed_on o
A8:
B1 is_closed_on o
by A2;
A9:
B2 is_closed_on o
by A4;
A10:
Den (
o,
U2) =
(Opers (U0,B2)) . o
by A5, MSUALG_1:def 6
.=
o /. B2
by Def8
.=
(Den (o,U0)) | (((B2 #) * the Arity of S) . o)
by A9, Def7
;
Den (
o,
U1) =
(Opers (U0,B1)) . o
by A3, MSUALG_1:def 6
.=
o /. B1
by Def8
.=
(Den (o,U0)) | (((B1 #) * the Arity of S) . o)
by A8, Def7
.=
(Den (o,U0)) | ((((B2 #) * the Arity of S) . o) /\ (((B1 #) * the Arity of S) . o))
by A1, Th2, XBOOLE_1:28
.=
(Den (o,U2)) | (((B1 #) * the Arity of S) . o)
by A10, RELAT_1:71
;
then
rng ((Den (o,U2)) | (((B1 #) * the Arity of S) . o)) c= Result (
o,
U1)
by RELAT_1:def 19;
then
rng ((Den (o,U2)) | (((B1 #) * the Arity of S) . o)) c= ( the Sorts of U1 * the ResultSort of S) . o
by MSUALG_1:def 5;
hence
B is_closed_on o
by A6;
verum
end;
hence
B is opers_closed
; the Charact of U1 = Opers (U2,B)
for x being object st x in the carrier' of S holds
the Charact of U1 . x = (Opers (U2,B)) . x
proof
let x be
object ;
( x in the carrier' of S implies the Charact of U1 . x = (Opers (U2,B)) . x )
assume
x in the
carrier' of
S
;
the Charact of U1 . x = (Opers (U2,B)) . x
then reconsider o =
x as
OperSymbol of
S ;
A11:
B1 is_closed_on o
by A2;
A12:
B2 is_closed_on o
by A4;
A13:
Den (
o,
U2) =
(Opers (U0,B2)) . o
by A5, MSUALG_1:def 6
.=
o /. B2
by Def8
.=
(Den (o,U0)) | (((B2 #) * the Arity of S) . o)
by A12, Def7
;
thus the
Charact of
U1 . x =
o /. B1
by A3, Def8
.=
(Den (o,U0)) | (((B1 #) * the Arity of S) . o)
by A11, Def7
.=
(Den (o,U0)) | ((((B2 #) * the Arity of S) . o) /\ (((B1 #) * the Arity of S) . o))
by A1, Th2, XBOOLE_1:28
.=
(Den (o,U2)) | (((B1 #) * the Arity of S) . o)
by A13, RELAT_1:71
.=
o /. B
by A6, A7, Def7
.=
(Opers (U2,B)) . x
by Def8
;
verum
end;
hence
the Charact of U1 = Opers (U2,B)
; verum