let X be non empty set ; :: thesis:
let vseq be sequence of (C_Normed_Algebra_of_BoundedFunctions X); :: thesis:
defpred S₁[ set , set ] means ex xseq being Complex_Sequence st
for n being Nat holds
( xseq . n = (modetrans ((vseq . n),X)) . $1 & xseq is convergent & $2 = lim xseq );
assume A1: vseq is CCauchy ; :: thesis:
A2: for x being Element of X ex y being Element of COMPLEX st S₁[x,y]

let x be Element of X; :: thesis:
deffunc H₁( Nat) -> Element of COMPLEX = (modetrans ((vseq . $1),X)) . x;
consider xseq being Complex_Sequence such that
A3: for n being Element of NAT holds xseq . n = H₁(n) from FUNCT_2:sch 4();
A4: for n being Nat holds xseq . n = H₁(n)

let n be Nat; :: thesis:
n in NAT by ORDINAL1:def 12;
hence xseq . n = H₁(n) by A3; :: thesis:

end;

reconsider y = lim xseq as Element of COMPLEX by XCMPLX_0:def 2;
take y ; :: thesis:
A5: for m, k being Nat holds |.((xseq . m) - (xseq . k)).| <= ||.((vseq . m) - (vseq . k)).||

proof

let m, k be Nat; :: thesis:
A6: ( m in NAT & k in NAT ) by ORDINAL1:def 12;
(vseq . m) - (vseq . k) in ComplexBoundedFunctions X ;
then consider h1 being Function of X,COMPLEX such that
A7: h1 = (vseq . m) - (vseq . k) and
A8: h1 | X is bounded ;
vseq . m in ComplexBoundedFunctions X ;
then ex vseqm being Function of X,COMPLEX st
( vseq . m = vseqm & vseqm | X is bounded ) ;
then A9: modetrans ((vseq . m),X) = vseq . m by Th12;
vseq . k in ComplexBoundedFunctions X ;
then ex vseqk being Function of X,COMPLEX st
( vseq . k = vseqk & vseqk | X is bounded ) ;
then A10: modetrans ((vseq . k),X) = vseq . k by Th12;
( xseq . m = (modetrans ((vseq . m),X)) . x & xseq . k = (modetrans ((vseq . k),X)) . x ) by A3, A6;
then (xseq . m) - (xseq . k) = h1 . x by A9, A10, A7, Th26;
hence |.((xseq . m) - (xseq . k)).| <= ||.((vseq . m) - (vseq . k)).|| by A7, A8, Th19; :: thesis:

end;

now :: thesis:

let e be Real; :: thesis:
assume e > 0 ; :: thesis:
then consider k being Nat such that
A11: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by A1, CSSPACE3:8;
reconsider k = k as Nat ;
take k = k; :: thesis:

hereby :: thesis:

let n be Nat; :: thesis:
assume n >= k ; :: thesis:
then A12: ||.((vseq . n) - (vseq . k)).|| < e by A11;
|.((xseq . n) - (xseq . k)).| <= ||.((vseq . n) - (vseq . k)).|| by A5;
hence |.((xseq . n) - (xseq . k)).| < e by A12, XXREAL_0:2; :: thesis:

end;

then xseq is convergent by COMSEQ_3:46;
hence S₁[x,y] by A4; :: thesis:

end;

consider tseq being Function of X,COMPLEX such that
A13: for x being Element of X holds S₁[x,tseq . x] from FUNCT_2:sch 3(A2);

now :: thesis:

let e1 be Real; :: thesis:
assume A14: e1 > 0 ; :: thesis:
reconsider e = e1 as Real ;
consider k being Nat such that
A15: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by A1, A14, CSSPACE3:8;
take k = k; :: thesis:

now :: thesis:

let m be Nat; :: thesis:
A16: ||.(vseq . m).|| = ||.vseq.|| . m by NORMSP_0:def 4;
assume m >= k ; :: thesis:
then A17: ||.((vseq . m) - (vseq . k)).|| < e by A15;
( |.(||.(vseq . m).|| - ||.(vseq . k).||).| <= ||.((vseq . m) - (vseq . k)).|| & ||.(vseq . k).|| = ||.vseq.|| . k ) by CLVECT_1:110, NORMSP_0:def 4;
hence |.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1 by A17, A16, XXREAL_0:2; :: thesis:

end;

hence for m being Nat st m >= k holds
|.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1 ; :: thesis:

end;

then A18: ||.vseq.|| is convergent by SEQ_4:41;

now :: thesis:

let x be set ; :: thesis:
assume A19: x in X /\ (dom tseq) ; :: thesis:
then consider xseq being Complex_Sequence such that
A20: for n being Nat holds xseq . n = (modetrans ((vseq . n),X)) . x and
A21: ( xseq is convergent & tseq . x = lim xseq ) by A13;
A22: for n being Nat holds |.xseq.| . n <= ||.vseq.|| . n

proof

let n be Nat; :: thesis:
A23: xseq . n = (modetrans ((vseq . n),X)) . x by A20;
vseq . n in ComplexBoundedFunctions X ;
then A24: ex h1 being Function of X,COMPLEX st
( vseq . n = h1 & h1 | X is bounded ) ;
then modetrans ((vseq . n),X) = vseq . n by Th12;
then |.(xseq . n).| <= ||.(vseq . n).|| by A19, A24, A23, Th19;
then |.xseq.| . n <= ||.(vseq . n).|| by VALUED_1:18;
hence |.xseq.| . n <= ||.vseq.|| . n by NORMSP_0:def 4; :: thesis:

end;

( |.xseq.| is convergent & |.(tseq . x).| = lim |.xseq.| ) by A21, SEQ_2:27;
hence |.(tseq . x).| <= lim ||.vseq.|| by A18, A22, SEQ_2:18; :: thesis:

end;

then for x being Element of X st x in X /\ (dom tseq) holds
|.(tseq /. x).| <= lim ||.vseq.|| ;
then tseq | X is bounded by CFUNCT_1:69;
then tseq in ComplexBoundedFunctions X ;
then reconsider tv = tseq as Point of (C_Normed_Algebra_of_BoundedFunctions X) ;
A25: for e being Real st e > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e

proof

let e be Real; :: thesis:
assume e > 0 ; :: thesis:
then consider k being Nat such that
A26: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by A1, CSSPACE3:8;
take k ; :: thesis:

now :: thesis:

let n be Nat; :: thesis:
assume A27: n >= k ; :: thesis:

now :: thesis:

let x be Element of X; :: thesis:
consider xseq being Complex_Sequence such that
A28: for n being Nat holds xseq . n = (modetrans ((vseq . n),X)) . x and
A29: xseq is convergent and
A30: tseq . x = lim xseq by A13;
reconsider xn = xseq . n as Element of COMPLEX by XCMPLX_0:def 2;
reconsider fseq = NAT --> xn as Complex_Sequence ;
set wseq = xseq - fseq;
deffunc H₁( Nat) -> object = |.((xseq . $1) - (xseq . n)).|;
consider rseq being Real_Sequence such that
A31: for m being Nat holds rseq . m = H₁(m) from SEQ_1:sch 1();
A32: for m, k being Nat holds |.((xseq . m) - (xseq . k)).| <= ||.((vseq . m) - (vseq . k)).||

proof

let m, k be Nat; :: thesis:
(vseq . m) - (vseq . k) in ComplexBoundedFunctions X ;
then consider h1 being Function of X,COMPLEX such that
A33: h1 = (vseq . m) - (vseq . k) and
A34: h1 | X is bounded ;
vseq . m in ComplexBoundedFunctions X ;
then ex vseqm being Function of X,COMPLEX st
( vseq . m = vseqm & vseqm | X is bounded ) ;
then A35: modetrans ((vseq . m),X) = vseq . m by Th12;
vseq . k in ComplexBoundedFunctions X ;
then ex vseqk being Function of X,COMPLEX st
( vseq . k = vseqk & vseqk | X is bounded ) ;
then A36: modetrans ((vseq . k),X) = vseq . k by Th12;
( xseq . m = (modetrans ((vseq . m),X)) . x & xseq . k = (modetrans ((vseq . k),X)) . x ) by A28;
then (xseq . m) - (xseq . k) = h1 . x by A35, A36, A33, Th26;
hence |.((xseq . m) - (xseq . k)).| <= ||.((vseq . m) - (vseq . k)).|| by A33, A34, Th19; :: thesis:

end;

A37: for m being Nat st m >= k holds
rseq . m <= e

proof

let m be Nat; :: thesis:
assume m >= k ; :: thesis:
then A38: ||.((vseq . n) - (vseq . m)).|| < e by A26, A27;
rseq . m = |.((xseq . m) - (xseq . n)).| by A31;
then rseq . m = |.((xseq . n) - (xseq . m)).| by COMPLEX1:60;
then rseq . m <= ||.((vseq . n) - (vseq . m)).|| by A32;
hence rseq . m <= e by A38, XXREAL_0:2; :: thesis:

end;

A39: now :: thesis:

let m be Element of NAT ; :: thesis:
(xseq - fseq) . m = (xseq . m) + ((- fseq) . m) by VALUED_1:1
.= (xseq . m) - (fseq . m) by VALUED_1:8 ;
hence (xseq - fseq) . m = (xseq . m) - (xseq . n) ; :: thesis:

end;

now :: thesis:

end;

then A40: rseq = |.(xseq - fseq).| by FUNCT_2:12;
A41: fseq is convergent by CFCONT_1:26;
A42: lim rseq <= e by A41, A37, A40, A29, RSSPACE2:5;
lim fseq = fseq . 0 by CFCONT_1:28;
then lim fseq = xseq . n ;
then lim (xseq - fseq) = (tseq . x) - (xseq . n) by A29, A30, A41, COMSEQ_2:26;
then lim rseq = |.((tseq . x) - (xseq . n)).| by A41, A29, A40, SEQ_2:27;
then |.((xseq . n) - (tseq . x)).| <= e by A42, COMPLEX1:60;
hence |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e by A28; :: thesis:

end;

hence for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e ; :: thesis:

end;

hence for n being Nat st n >= k holds
for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e ; :: thesis:

end;

A43: for e being Real st e > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e

proof

let e be Real; :: thesis:
assume e > 0 ; :: thesis:
then consider k being Nat such that
A44: for n being Nat st n >= k holds
for x being Element of X holds |.(((modetrans ((vseq . n),X)) . x) - (tseq . x)).| <= e by A25;
take k ; :: thesis:

hereby :: thesis:

let n be Nat; :: thesis:
assume A45: n >= k ; :: thesis:
vseq . n in ComplexBoundedFunctions X ;
then consider f1 being Function of X,COMPLEX such that
A46: f1 = vseq . n and
f1 | X is bounded ;
(vseq . n) - tv in ComplexBoundedFunctions X ;
then consider h1 being Function of X,COMPLEX such that
A47: h1 = (vseq . n) - tv and
A48: h1 | X is bounded ;

A49: now :: thesis:

let t be Element of X; :: thesis:
( modetrans ((vseq . n),X) = f1 & h1 . t = (f1 . t) - (tseq . t) ) by A46, A47, Def7, Th26;
hence |.(h1 . t).| <= e by A44, A45; :: thesis:

end;

A50: now :: thesis:

let r be Real; :: thesis:
assume r in PreNorms h1 ; :: thesis:
then ex t being Element of X st r = |.(h1 . t).| ;
hence r <= e by A49; :: thesis:

end;

( ( for s being Real st s in PreNorms h1 holds
s <= e ) implies upper_bound (PreNorms h1) <= e ) by SEQ_4:45;
hence ||.((vseq . n) - tv).|| <= e by A47, A48, A50, Th13; :: thesis:

end;

for e being Real st e > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e

proof

let e be Real; :: thesis:
assume A51: e > 0 ; :: thesis:
consider m being Nat such that
A52: for n being Nat st n >= m holds
||.((vseq . n) - tv).|| <= e / 2 by A43, A51;
take m ; :: thesis:
A53: e / 2 < e by A51, XREAL_1:216;

hereby :: thesis:

let n be Nat; :: thesis:
assume n >= m ; :: thesis:
then ||.((vseq . n) - tv).|| <= e / 2 by A52;
hence ||.((vseq . n) - tv).|| < e by A53, XXREAL_0:2; :: thesis:

end;

hence vseq is convergent ; :: thesis: