let X, Y be RealNormSpace; :: thesis:
assume A1: Y is complete ; :: thesis:
let vseq be sequence of (R_NormSpace_of_BoundedLinearOperators (X,Y)); :: thesis:
assume A2: vseq is Cauchy_sequence_by_Norm ; :: thesis:
defpred S₁[ set , set ] means ex xseq being sequence of Y st
( ( for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . $1 ) & xseq is convergent & $2 = lim xseq );
A3: for x being Element of X ex y being Element of Y st S₁[x,y]

let x be Element of X; :: thesis:
deffunc H₁( Nat) -> Element of the carrier of Y = (modetrans ((vseq . $1),X,Y)) . x;
consider xseq being sequence of Y such that
A4: for n being Element of NAT holds xseq . n = H₁(n) from FUNCT_2:sch 4();
A5: 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 A4; :: thesis:

end;

take lim xseq ; :: thesis:
A6: for m, k being Nat holds ||.((xseq . m) - (xseq . k)).|| <= ||.((vseq . m) - (vseq . k)).|| * ||.x.||

proof

let m, k be Nat; :: thesis:
reconsider h1 = (vseq . m) - (vseq . k) as Lipschitzian LinearOperator of X,Y by Def9;
k in NAT by ORDINAL1:def 12;
then A7: xseq . k = (modetrans ((vseq . k),X,Y)) . x by A4;
vseq . m is Lipschitzian LinearOperator of X,Y by Def9;
then A8: modetrans ((vseq . m),X,Y) = vseq . m by Th29;
vseq . k is Lipschitzian LinearOperator of X,Y by Def9;
then A9: modetrans ((vseq . k),X,Y) = vseq . k by Th29;
m in NAT by ORDINAL1:def 12;
then xseq . m = (modetrans ((vseq . m),X,Y)) . x by A4;
then (xseq . m) - (xseq . k) = h1 . x by A8, A9, A7, Th40;
hence ||.((xseq . m) - (xseq . k)).|| <= ||.((vseq . m) - (vseq . k)).|| * ||.x.|| by Th32; :: thesis:

end;

now :: thesis:

let e be Real; :: thesis:
assume A10: e > 0 ; :: thesis:

now :: thesis:

per cases ;

case A11: x = 0. X ; :: thesis:

reconsider k = 0 as Nat ;
take k = k; :: thesis:
thus for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e :: thesis:

proof

let n, m be Nat; :: thesis:
assume that
n >= k and
m >= k ; :: thesis:
m in NAT by ORDINAL1:def 12;
then A12: xseq . m = (modetrans ((vseq . m),X,Y)) . x by A4
.= (modetrans ((vseq . m),X,Y)) . (0 * x) by A11
.= 0 * ((modetrans ((vseq . m),X,Y)) . x) by Def5
.= 0. Y by RLVECT_1:10 ;
n in NAT by ORDINAL1:def 12;
then xseq . n = (modetrans ((vseq . n),X,Y)) . x by A4
.= (modetrans ((vseq . n),X,Y)) . (0 * x) by A11
.= 0 * ((modetrans ((vseq . n),X,Y)) . x) by Def5
.= 0. Y by RLVECT_1:10 ;
then ||.((xseq . n) - (xseq . m)).|| = ||.(0. Y).|| by A12
.= 0 ;
hence ||.((xseq . n) - (xseq . m)).|| < e by A10; :: thesis:

end;

case x <> 0. X ; :: thesis:

then A13: ||.x.|| <> 0 by NORMSP_0:def 5;
then A14: ||.x.|| > 0 ;
then e / ||.x.|| > 0 by A10, XREAL_1:139;
then consider k being Nat such that
A15: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e / ||.x.|| by A2, RSSPACE3:8;
take k = k; :: thesis:
thus for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e :: thesis:

proof

let n, m be Nat; :: thesis:
assume that
A16: n >= k and
A17: m >= k ; :: thesis:
||.((vseq . n) - (vseq . m)).|| < e / ||.x.|| by A15, A16, A17;
then A18: ||.((vseq . n) - (vseq . m)).|| * ||.x.|| < (e / ||.x.||) * ||.x.|| by A14, XREAL_1:68;
A19: (e / ||.x.||) * ||.x.|| = (e * (||.x.|| ")) * ||.x.|| by XCMPLX_0:def 9
.= e * ((||.x.|| ") * ||.x.||)
.= e * 1 by A13, XCMPLX_0:def 7
.= e ;
||.((xseq . n) - (xseq . m)).|| <= ||.((vseq . n) - (vseq . m)).|| * ||.x.|| by A6;
hence ||.((xseq . n) - (xseq . m)).|| < e by A18, A19, XXREAL_0:2; :: thesis:

end;

hence ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((xseq . n) - (xseq . m)).|| < e ; :: thesis:

end;

then xseq is Cauchy_sequence_by_Norm by RSSPACE3:8;
then xseq is convergent by A1;
hence S₁[x, lim xseq] by A5; :: thesis:

end;

consider f being Function of the carrier of X, the carrier of Y such that
A20: for x being Element of X holds S₁[x,f . x] from FUNCT_2:sch 3(A3);
reconsider tseq = f as Function of X,Y ;

A21: now :: thesis:

let x, y be VECTOR of X; :: thesis:
consider xseq being sequence of Y such that
A22: for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . x and
A23: xseq is convergent and
A24: tseq . x = lim xseq by A20;
consider zseq being sequence of Y such that
A25: for n being Nat holds zseq . n = (modetrans ((vseq . n),X,Y)) . (x + y) and
zseq is convergent and
A26: tseq . (x + y) = lim zseq by A20;
consider yseq being sequence of Y such that
A27: for n being Nat holds yseq . n = (modetrans ((vseq . n),X,Y)) . y and
A28: yseq is convergent and
A29: tseq . y = lim yseq by A20;

now :: thesis:

let n be Nat; :: thesis:
thus zseq . n = (modetrans ((vseq . n),X,Y)) . (x + y) by A25
.= ((modetrans ((vseq . n),X,Y)) . x) + ((modetrans ((vseq . n),X,Y)) . y) by VECTSP_1:def 20
.= (xseq . n) + ((modetrans ((vseq . n),X,Y)) . y) by A22
.= (xseq . n) + (yseq . n) by A27 ; :: thesis:

end;

then zseq = xseq + yseq by NORMSP_1:def 2;
hence tseq . (x + y) = (tseq . x) + (tseq . y) by A23, A24, A28, A29, A26, NORMSP_1:25; :: thesis:

end;

now :: thesis:

let x be VECTOR of X; :: thesis:
let a be Real; :: thesis:
consider xseq being sequence of Y such that
A30: for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . x and
A31: xseq is convergent and
A32: tseq . x = lim xseq by A20;
consider zseq being sequence of Y such that
A33: for n being Nat holds zseq . n = (modetrans ((vseq . n),X,Y)) . (a * x) and
zseq is convergent and
A34: tseq . (a * x) = lim zseq by A20;

now :: thesis:

let n be Nat; :: thesis:
thus zseq . n = (modetrans ((vseq . n),X,Y)) . (a * x) by A33
.= a * ((modetrans ((vseq . n),X,Y)) . x) by Def5
.= a * (xseq . n) by A30 ; :: thesis:

end;

then zseq = a * xseq by NORMSP_1:def 5;
hence tseq . (a * x) = a * (tseq . x) by A31, A32, A34, NORMSP_1:28; :: thesis:

end;

then reconsider tseq = tseq as LinearOperator of X,Y by A21, Def5, VECTSP_1:def 20;

now :: thesis:

let e1 be Real; :: thesis:
assume A35: e1 > 0 ; :: thesis:
reconsider e = e1 as Real ;
consider k being Nat such that
A36: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by A2, A35, RSSPACE3:8;
reconsider k = k as Nat ;
take k = k; :: thesis:

now :: thesis:

let m be Nat; :: thesis:
assume m >= k ; :: thesis:
then A37: ||.((vseq . m) - (vseq . k)).|| < e by A36;
A38: ||.(vseq . m).|| = ||.vseq.|| . m by NORMSP_0:def 4;
A39: ||.(vseq . k).|| = ||.vseq.|| . k by NORMSP_0:def 4;
|.(||.(vseq . m).|| - ||.(vseq . k).||).| <= ||.((vseq . m) - (vseq . k)).|| by NORMSP_1:9;
hence |.((||.vseq.|| . m) - (||.vseq.|| . k)).| < e1 by A39, A38, A37, XXREAL_0:2; :: thesis:

end;

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

end;

then A40: ||.vseq.|| is convergent by SEQ_4:41;
A41: tseq is Lipschitzian

proof

take lim ||.vseq.|| ; :: according to LOPBAN_1:def 8 :: thesis:

A42: now :: thesis:

let x be VECTOR of X; :: thesis:
consider xseq being sequence of Y such that
A43: for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . x and
A44: xseq is convergent and
A45: tseq . x = lim xseq by A20;
A46: ||.(tseq . x).|| = lim ||.xseq.|| by A44, A45, Th20;
A47: for m being Nat holds ||.(xseq . m).|| <= ||.(vseq . m).|| * ||.x.||

proof

let m be Nat; :: thesis:
A48: xseq . m = (modetrans ((vseq . m),X,Y)) . x by A43;
vseq . m is Lipschitzian LinearOperator of X,Y by Def9;
hence ||.(xseq . m).|| <= ||.(vseq . m).|| * ||.x.|| by A48, Th29, Th32; :: thesis:

end;

A49: for n being Nat holds ||.xseq.|| . n <= (||.x.|| (#) ||.vseq.||) . n

proof

let n be Nat; :: thesis:
A50: ||.xseq.|| . n = ||.(xseq . n).|| by NORMSP_0:def 4;
A51: ||.(vseq . n).|| = ||.vseq.|| . n by NORMSP_0:def 4;
||.(xseq . n).|| <= ||.(vseq . n).|| * ||.x.|| by A47;
hence ||.xseq.|| . n <= (||.x.|| (#) ||.vseq.||) . n by A50, A51, SEQ_1:9; :: thesis:

end;

A52: ||.x.|| (#) ||.vseq.|| is convergent by A40;
A53: lim (||.x.|| (#) ||.vseq.||) = (lim ||.vseq.||) * ||.x.|| by A40, SEQ_2:8;
||.xseq.|| is convergent by A44, A45, Th20;
hence ||.(tseq . x).|| <= (lim ||.vseq.||) * ||.x.|| by A46, A49, A52, A53, SEQ_2:18; :: thesis:

end;

now :: thesis:

let n be Nat; :: thesis:
||.(vseq . n).|| >= 0 ;
hence ||.vseq.|| . n >= 0 by NORMSP_0:def 4; :: thesis:

end;

hence ( 0 <= lim ||.vseq.|| & ( for x being VECTOR of X holds ||.(tseq . x).|| <= (lim ||.vseq.||) * ||.x.|| ) ) by A40, A42, SEQ_2:17; :: thesis:

end;

A54: for e being Real st e > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
for x being VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * ||.x.||

proof

let e be Real; :: thesis:
assume e > 0 ; :: thesis:
then consider k being Nat such that
A55: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by A2, RSSPACE3:8;
take k ; :: thesis:

now :: thesis:

let n be Nat; :: thesis:
assume A56: n >= k ; :: thesis:

now :: thesis:

let x be VECTOR of X; :: thesis:
consider xseq being sequence of Y such that
A57: for n being Nat holds xseq . n = (modetrans ((vseq . n),X,Y)) . x and
A58: xseq is convergent and
A59: tseq . x = lim xseq by A20;
A60: for m, k being Nat holds ||.((xseq . m) - (xseq . k)).|| <= ||.((vseq . m) - (vseq . k)).|| * ||.x.||

proof

let m, k be Nat; :: thesis:
reconsider h1 = (vseq . m) - (vseq . k) as Lipschitzian LinearOperator of X,Y by Def9;
A61: xseq . k = (modetrans ((vseq . k),X,Y)) . x by A57;
vseq . m is Lipschitzian LinearOperator of X,Y by Def9;
then A62: modetrans ((vseq . m),X,Y) = vseq . m by Th29;
vseq . k is Lipschitzian LinearOperator of X,Y by Def9;
then A63: modetrans ((vseq . k),X,Y) = vseq . k by Th29;
xseq . m = (modetrans ((vseq . m),X,Y)) . x by A57;
then (xseq . m) - (xseq . k) = h1 . x by A62, A63, A61, Th40;
hence ||.((xseq . m) - (xseq . k)).|| <= ||.((vseq . m) - (vseq . k)).|| * ||.x.|| by Th32; :: thesis:

end;

A64: for m being Nat st m >= k holds
||.((xseq . n) - (xseq . m)).|| <= e * ||.x.||

proof

let m be Nat; :: thesis:
assume m >= k ; :: thesis:
then A65: ||.((vseq . n) - (vseq . m)).|| < e by A55, A56;
A66: ||.((xseq . n) - (xseq . m)).|| <= ||.((vseq . n) - (vseq . m)).|| * ||.x.|| by A60;
||.((vseq . n) - (vseq . m)).|| * ||.x.|| <= e * ||.x.|| by A65, XREAL_1:64;
hence ||.((xseq . n) - (xseq . m)).|| <= e * ||.x.|| by A66, XXREAL_0:2; :: thesis:

end;

||.((xseq . n) - (tseq . x)).|| <= e * ||.x.||

proof

deffunc H₁( Nat) -> object = ||.((xseq . $1) - (xseq . n)).||;
consider rseq being Real_Sequence such that
A67: for m being Nat holds rseq . m = H₁(m) from SEQ_1:sch 1();

now :: thesis:

let x be object ; :: thesis:
assume x in NAT ; :: thesis:
then reconsider k = x as Nat ;
thus rseq . x = ||.((xseq . k) - (xseq . n)).|| by A67
.= ||.((xseq - (xseq . n)) . k).|| by NORMSP_1:def 4
.= ||.(xseq - (xseq . n)).|| . x by NORMSP_0:def 4 ; :: thesis:

end;

then A68: rseq = ||.(xseq - (xseq . n)).|| by FUNCT_2:12;
A69: xseq - (xseq . n) is convergent by A58, NORMSP_1:21;
lim (xseq - (xseq . n)) = (tseq . x) - (xseq . n) by A58, A59, NORMSP_1:27;
then A70: lim rseq = ||.((tseq . x) - (xseq . n)).|| by A69, A68, Th41;
for m being Nat st m >= k holds
rseq . m <= e * ||.x.||

proof

let m be Nat; :: thesis:
assume A71: m >= k ; :: thesis:
rseq . m = ||.((xseq . m) - (xseq . n)).|| by A67
.= ||.((xseq . n) - (xseq . m)).|| by NORMSP_1:7 ;
hence rseq . m <= e * ||.x.|| by A64, A71; :: thesis:

end;

then lim rseq <= e * ||.x.|| by A69, A68, Lm3, Th41;
hence ||.((xseq . n) - (tseq . x)).|| <= e * ||.x.|| by A70, NORMSP_1:7; :: thesis:

end;

hence ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * ||.x.|| by A57; :: thesis:

end;

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

end;

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

end;

reconsider tseq = tseq as Lipschitzian LinearOperator of X,Y by A41;
reconsider tv = tseq as Point of (R_NormSpace_of_BoundedLinearOperators (X,Y)) by Def9;
A72: 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 A73: e > 0 ; :: thesis:
consider k being Nat such that
A74: for n being Nat st n >= k holds
for x being VECTOR of X holds ||.(((modetrans ((vseq . n),X,Y)) . x) - (tseq . x)).|| <= e * ||.x.|| by A54, A73;

now :: thesis:

set g1 = tseq;
let n be Nat; :: thesis:
assume A75: n >= k ; :: thesis:
reconsider h1 = (vseq . n) - tv as Lipschitzian LinearOperator of X,Y by Def9;
set f1 = modetrans ((vseq . n),X,Y);

A76: now :: thesis:

let t be VECTOR of X; :: thesis:
assume ||.t.|| <= 1 ; :: thesis:
then A77: e * ||.t.|| <= e * 1 by A73, XREAL_1:64;
A78: ||.(((modetrans ((vseq . n),X,Y)) . t) - (tseq . t)).|| <= e * ||.t.|| by A74, A75;
vseq . n is Lipschitzian LinearOperator of X,Y by Def9;
then modetrans ((vseq . n),X,Y) = vseq . n by Th29;
then ||.(h1 . t).|| = ||.(((modetrans ((vseq . n),X,Y)) . t) - (tseq . t)).|| by Th40;
hence ||.(h1 . t).|| <= e by A78, A77, XXREAL_0:2; :: thesis:

end;

A79: now :: thesis:

let r be Real; :: thesis:
assume r in PreNorms h1 ; :: thesis:
then ex t being VECTOR of X st
( r = ||.(h1 . t).|| & ||.t.|| <= 1 ) ;
hence r <= e by A76; :: thesis:

end;

A80: ( ( for s being Real st s in PreNorms h1 holds
s <= e ) implies upper_bound (PreNorms h1) <= e ) by SEQ_4:45;
(BoundedLinearOperatorsNorm (X,Y)) . ((vseq . n) - tv) = upper_bound (PreNorms h1) by Th30;
hence ||.((vseq . n) - tv).|| <= e by A79, A80; :: thesis:

end;

hence ex k being Nat st
for n being Nat st n >= k holds
||.((vseq . n) - tv).|| <= e ; :: 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 A81: e > 0 ; :: thesis:
consider m being Nat such that
A82: for n being Nat st n >= m holds
||.((vseq . n) - tv).|| <= e / 2 by A72, A81, XREAL_1:215;
A83: e / 2 < e by A81, XREAL_1:216;

now :: thesis:

let n be Nat; :: thesis:
assume n >= m ; :: thesis:
then ||.((vseq . n) - tv).|| <= e / 2 by A82;
hence ||.((vseq . n) - tv).|| < e by A83, XXREAL_0:2; :: thesis:

end;

hence ex m being Nat st
for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e ; :: thesis:

end;

hence vseq is convergent by NORMSP_1:def 6; :: thesis: