let X, Y, Z be RealNormSpace; :: thesis:
set XC = the carrier of X;
set YC = the carrier of Y;
set ZC = the carrier of Z;
consider I0 being Function of (Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)))),(Funcs ([: the carrier of X, the carrier of Y:], the carrier of Z)) such that
A1: ( I0 is bijective & ( for f being Function of the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))
for d, e being object st d in the carrier of X & e in the carrier of Y holds
(I0 . f) . (d,e) = (f . d) . e ) ) by NDIFF_6:1;
set LXYZ = the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z))));
set BXYZ = the carrier of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z));
set LYZ = the carrier of (R_NormSpace_of_BoundedLinearOperators (Y,Z));
A3: the carrier of (R_NormSpace_of_BoundedLinearOperators (Y,Z)) c= Funcs ( the carrier of Y, the carrier of Z) by XBOOLE_1:1;
A4: the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))) c= Funcs ( the carrier of X, the carrier of (R_NormSpace_of_BoundedLinearOperators (Y,Z))) by XBOOLE_1:1;
Funcs ( the carrier of X, the carrier of (R_NormSpace_of_BoundedLinearOperators (Y,Z))) c= Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)))

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in Funcs ( the carrier of X, the carrier of (R_NormSpace_of_BoundedLinearOperators (Y,Z))) ; :: thesis:
then consider f being Function such that
A6: ( x = f & dom f = the carrier of X & rng f c= the carrier of (R_NormSpace_of_BoundedLinearOperators (Y,Z)) ) by FUNCT_2:def 2;
rng f c= Funcs ( the carrier of Y, the carrier of Z) by A3, A6, XBOOLE_1:1;
hence x in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))) by A6, FUNCT_2:def 2; :: thesis:

end;

proof

let f be Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))); :: thesis:
A9: I . f = I0 . f by FUNCT_1:49;
A10: f in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))) by A7, TARSKI:def 3;
A11: f is Function of the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)) by A7, TARSKI:def 3, FUNCT_2:66;
reconsider g = f as Function of the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)) by A7, TARSKI:def 3, FUNCT_2:66;
reconsider F = f as Lipschitzian LinearOperator of X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)) by LOPBAN_1:def 9;
thus for x being Point of X
for y being Point of Y ex G being Lipschitzian LinearOperator of Y,Z st
( G = f . x & (I . f) . (x,y) = G . y ) :: thesis:

proof

let x be Point of X; :: thesis:
let y be Point of Y; :: thesis:
g . x = F . x ;
then reconsider G = g . x as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
take G ; :: thesis:
thus ( G = f . x & (I . f) . (x,y) = G . y ) by A1, A9; :: thesis:

end;

reconsider BL = I0 . f as Function of [:X,Y:],Z by A10, FUNCT_2:5, FUNCT_2:66;
A15: for x1, x2 being Point of X
for y being Point of Y holds BL . ((x1 + x2),y) = (BL . (x1,y)) + (BL . (x2,y))

proof

let x1, x2 be Point of X; :: thesis:
let y be Point of Y; :: thesis:
A16: BL . (x1,y) = (F . x1) . y by A1, A11;
A17: BL . (x2,y) = (F . x2) . y by A1, A11;
A18: BL . ((x1 + x2),y) = (F . (x1 + x2)) . y by A1, A11;
F . (x1 + x2) = (F . x1) + (F . x2) by VECTSP_1:def 20;
hence BL . ((x1 + x2),y) = (BL . (x1,y)) + (BL . (x2,y)) by A16, A17, A18, LOPBAN_1:35; :: thesis:

end;

A19: for x being Point of X
for y being Point of Y
for a being Real holds BL . ((a * x),y) = a * (BL . (x,y))

proof

let x be Point of X; :: thesis:
let y be Point of Y; :: thesis:
let a be Real; :: thesis:
A20: BL . ((a * x),y) = (F . (a * x)) . y by A1, A11;
A21: BL . (x,y) = (F . x) . y by A1, A11;
F . (a * x) = a * (F . x) by LOPBAN_1:def 5;
hence BL . ((a * x),y) = a * (BL . (x,y)) by A20, A21, LOPBAN_1:36; :: thesis:

end;

A22: for x being Point of X
for y1, y2 being Point of Y holds BL . (x,(y1 + y2)) = (BL . (x,y1)) + (BL . (x,y2))

proof

let x be Point of X; :: thesis:
let y1, y2 be Point of Y; :: thesis:
reconsider Fx = F . x as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
A23: BL . (x,y1) = Fx . y1 by A1, A11;
A24: BL . (x,y2) = Fx . y2 by A1, A11;
BL . (x,(y1 + y2)) = Fx . (y1 + y2) by A1, A11;
hence BL . (x,(y1 + y2)) = (BL . (x,y1)) + (BL . (x,y2)) by A23, A24, VECTSP_1:def 20; :: thesis:

end;

A26: for x being Point of X
for y being Point of Y
for a being Real holds BL . (x,(a * y)) = a * (BL . (x,y))

proof

let x be Point of X; :: thesis:
let y be Point of Y; :: thesis:
let a be Real; :: thesis:
reconsider Fx = F . x as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
A27: BL . (x,y) = Fx . y by A1, A11;
BL . (x,(a * y)) = Fx . (a * y) by A1, A11;
hence BL . (x,(a * y)) = a * (BL . (x,y)) by A27, LOPBAN_1:def 5; :: thesis:

end;

reconsider BL = BL as BilinearOperator of X,Y,Z by A15, A19, A22, A26, LOPBAN_8:12;
A29: for x being Point of X
for y being Point of Y holds ||.(BL . (x,y)).|| <= (||.f.|| * ||.x.||) * ||.y.||

proof

let x be Point of X; :: thesis:
let y be Point of Y; :: thesis:
reconsider Fx = F . x as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
A30: BL . (x,y) = Fx . y by A1, A11;
A31: ||.(Fx . y).|| <= ||.(F . x).|| * ||.y.|| by LOPBAN_1:32;
||.(F . x).|| * ||.y.|| <= (||.f.|| * ||.x.||) * ||.y.|| by LOPBAN_1:32, XREAL_1:64;
hence ||.(BL . (x,y)).|| <= (||.f.|| * ||.x.||) * ||.y.|| by A30, A31, XXREAL_0:2; :: thesis:

end;

then reconsider BL = BL as Lipschitzian BilinearOperator of X,Y,Z by Def8;
reconsider BLp = BL as Point of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) by Def9;
I . f = BL by FUNCT_1:49;
hence I . f is Lipschitzian BilinearOperator of X,Y,Z ; :: thesis:
BLp in the carrier of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) ;
hence I . f in the carrier of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) by FUNCT_1:49; :: thesis:
A33: ||.BLp.|| = upper_bound (PreNorms (modetrans (BL,X,Y,Z))) by Def13
.= upper_bound (PreNorms BL) ;

now :: thesis:

let r be Real; :: thesis:
assume r in PreNorms BL ; :: thesis:
then consider t being VECTOR of X, s being VECTOR of Y such that
A34: r = ||.(BL . (t,s)).|| and
A35: ( ||.t.|| <= 1 & ||.s.|| <= 1 ) ;
A36: ||.(BL . (t,s)).|| <= (||.f.|| * ||.t.||) * ||.s.|| by A29;
||.t.|| * ||.s.|| <= 1 * 1 by A35, XREAL_1:66;
then ||.f.|| * (||.t.|| * ||.s.||) <= ||.f.|| * 1 by XREAL_1:64;
hence r <= ||.f.|| by A34, A36, XXREAL_0:2; :: thesis:

end;

then A37: ||.BLp.|| <= ||.f.|| by A33, SEQ_4:45;
A39: ||.f.|| = upper_bound (PreNorms (modetrans (F,X,(R_NormSpace_of_BoundedLinearOperators (Y,Z))))) by LOPBAN_1:def 13
.= upper_bound (PreNorms F) by LOPBAN_1:29 ;

now :: thesis:

let r be Real; :: thesis:
assume r in PreNorms F ; :: thesis:
then consider x being VECTOR of X such that
A40: r = ||.(F . x).|| and
A41: ||.x.|| <= 1 ;
reconsider Fx = F . x as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
A42: ||.(F . x).|| = upper_bound (PreNorms Fx) by LOPBAN_1:30;

now :: thesis:

let s be Real; :: thesis:
assume s in PreNorms Fx ; :: thesis:
then consider y being Point of Y such that
B42: s = ||.(Fx . y).|| and
A43: ||.y.|| <= 1 ;
A44: ||.(Fx . y).|| = ||.(BL . (x,y)).|| by A1, A11;
A45: ||.(BL . (x,y)).|| <= (||.BLp.|| * ||.x.||) * ||.y.|| by Th32;
||.x.|| * ||.y.|| <= 1 * 1 by A41, A43, XREAL_1:66;
then ||.BLp.|| * (||.x.|| * ||.y.||) <= ||.BLp.|| * 1 by XREAL_1:64;
hence s <= ||.BLp.|| by B42, A44, A45, XXREAL_0:2; :: thesis:

end;

hence r <= ||.BLp.|| by A40, A42, SEQ_4:45; :: thesis:

end;

then A46: ||.f.|| <= ||.BLp.|| by A39, SEQ_4:45;
take BLp ; :: thesis:
thus ( BLp = I . f & ||.f.|| = ||.BLp.|| ) by A37, A46, FUNCT_1:49, XXREAL_0:1; :: thesis:

end;

then rng I c= the carrier of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) by FUNCT_2:114;
then reconsider I = I as Function of the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))), the carrier of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) by FUNCT_2:6;
A47: for x1, x2 being Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))) holds I . (x1 + x2) = (I . x1) + (I . x2)

proof

let x1, x2 be Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))); :: thesis:
for p being Point of X
for q being Point of Y holds (I . (x1 + x2)) . (p,q) = ((I . x1) . (p,q)) + ((I . x2) . (p,q))

proof

let p be Point of X; :: thesis:
let q be Point of Y; :: thesis:
consider Gx1p being Lipschitzian LinearOperator of Y,Z such that
A48: ( Gx1p = x1 . p & (I . x1) . (p,q) = Gx1p . q ) by A8;
consider Gx2p being Lipschitzian LinearOperator of Y,Z such that
A49: ( Gx2p = x2 . p & (I . x2) . (p,q) = Gx2p . q ) by A8;
consider Gx1x2p being Lipschitzian LinearOperator of Y,Z such that
A50: ( Gx1x2p = (x1 + x2) . p & (I . (x1 + x2)) . (p,q) = Gx1x2p . q ) by A8;
(x1 + x2) . p = (x1 . p) + (x2 . p) by LOPBAN_1:35;
hence (I . (x1 + x2)) . (p,q) = ((I . x1) . (p,q)) + ((I . x2) . (p,q)) by A48, A49, A50, LOPBAN_1:35; :: thesis:

end;

hence I . (x1 + x2) = (I . x1) + (I . x2) by Th35; :: thesis:

end;

for x being Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z))))
for a being Real holds I . (a * x) = a * (I . x)

proof

let x be Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))); :: thesis:
let a be Real; :: thesis:
for p being Point of X
for q being Point of Y holds (I . (a * x)) . (p,q) = a * ((I . x) . (p,q))

proof

let p be Point of X; :: thesis:
let q be Point of Y; :: thesis:
consider Gxp being Lipschitzian LinearOperator of Y,Z such that
A52: ( Gxp = x . p & (I . x) . (p,q) = Gxp . q ) by A8;
consider Gxap being Lipschitzian LinearOperator of Y,Z such that
A53: ( Gxap = (a * x) . p & (I . (a * x)) . (p,q) = Gxap . q ) by A8;
(a * x) . p = a * (x . p) by LOPBAN_1:36;
hence (I . (a * x)) . (p,q) = a * ((I . x) . (p,q)) by A52, A53, LOPBAN_1:36; :: thesis:

end;

hence I . (a * x) = a * (I . x) by Th36; :: thesis:

end;

then reconsider I = I as LinearOperator of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))),(R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) by A47, LOPBAN_1:def 5, VECTSP_1:def 20;
A55: for u being Point of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))) holds
( ||.u.|| = ||.(I . u).|| & ( for x being Point of X
for y being Point of Y holds (I . u) . (x,y) = (u . x) . y ) )

proof

let u be Point of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))); :: thesis:
consider Iu being Point of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) such that
A56: ( Iu = I . u & ||.u.|| = ||.Iu.|| ) by A8;
thus ||.u.|| = ||.(I . u).|| by A56; :: thesis:
let p be Point of X; :: thesis:
let q be Point of Y; :: thesis:
consider G being Lipschitzian LinearOperator of Y,Z such that
A57: ( G = u . p & (I . u) . (p,q) = G . q ) by A8;
thus (I . u) . (p,q) = (u . p) . q by A57; :: thesis:

end;

for y being object st y in the carrier of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) holds
ex x being object st
( x in the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))) & y = I . x )

proof

let y be object ; :: thesis:
assume A59: y in the carrier of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) ; :: thesis:
then y in Funcs ([: the carrier of X, the carrier of Y:], the carrier of Z) by TARSKI:def 3;
then y in rng I0 by A1, FUNCT_2:def 3;
then consider f being object such that
A60: ( f in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))) & I0 . f = y ) by FUNCT_2:11;
reconsider f = f as Function of the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)) by A60, FUNCT_2:66;
reconsider BL = y as Lipschitzian BilinearOperator of X,Y,Z by A59, Def9;
reconsider BLp = BL as Point of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) by Def9;
A62: dom f = the carrier of X by FUNCT_2:def 1;
for x being object st x in the carrier of X holds
f . x in the carrier of (R_NormSpace_of_BoundedLinearOperators (Y,Z))

proof

let x be object ; :: thesis:
assume A63: x in the carrier of X ; :: thesis:
then reconsider fx = f . x as Function of the carrier of Y, the carrier of Z by FUNCT_2:5, FUNCT_2:66;
reconsider xp = x as Point of X by A63;
A64: for p, q being Point of Y holds fx . (p + q) = (fx . p) + (fx . q)

proof

let p, q be Point of Y; :: thesis:
A65: BL . (xp,p) = fx . p by A60, A1;
A66: BL . (xp,q) = fx . q by A60, A1;
BL . (xp,(p + q)) = fx . (p + q) by A60, A1;
hence fx . (p + q) = (fx . p) + (fx . q) by A65, A66, LOPBAN_8:12; :: thesis:

end;

for p being Point of Y
for a being Real holds fx . (a * p) = a * (fx . p)

proof

let p be Point of Y; :: thesis:
let a be Real; :: thesis:
A68: BL . (xp,p) = fx . p by A60, A1;
BL . (xp,(a * p)) = fx . (a * p) by A60, A1;
hence fx . (a * p) = a * (fx . p) by A68, LOPBAN_8:12; :: thesis:

end;

then reconsider fx = fx as LinearOperator of Y,Z by A64, LOPBAN_1:def 5, VECTSP_1:def 20;
for p being Point of Y holds ||.(fx . p).|| <= (||.BLp.|| * ||.xp.||) * ||.p.||

proof

let p be Point of Y; :: thesis:
BL . (xp,p) = fx . p by A60, A1;
hence ||.(fx . p).|| <= (||.BLp.|| * ||.xp.||) * ||.p.|| by Th32; :: thesis:

end;

then reconsider fx = fx as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 8;
fx in the carrier of (R_NormSpace_of_BoundedLinearOperators (Y,Z)) by LOPBAN_1:def 9;
hence f . x in the carrier of (R_NormSpace_of_BoundedLinearOperators (Y,Z)) ; :: thesis:

end;

then reconsider f = f as Function of the carrier of X, the carrier of (R_NormSpace_of_BoundedLinearOperators (Y,Z)) by A62, FUNCT_2:3;
A71: for x1, x2 being Point of X holds f . (x1 + x2) = (f . x1) + (f . x2)

proof

let x1, x2 be Point of X; :: thesis:
reconsider fx1x2 = f . (x1 + x2) as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
reconsider fx1 = f . x1 as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
reconsider fx2 = f . x2 as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
for y being Point of Y holds fx1x2 . y = (fx1 . y) + (fx2 . y)

proof

let y be Point of Y; :: thesis:
A72: BL . (x1,y) = fx1 . y by A60, A1;
A73: BL . (x2,y) = fx2 . y by A60, A1;
BL . ((x1 + x2),y) = fx1x2 . y by A60, A1;
hence fx1x2 . y = (fx1 . y) + (fx2 . y) by A72, A73, LOPBAN_8:12; :: thesis:

end;

hence f . (x1 + x2) = (f . x1) + (f . x2) by LOPBAN_1:35; :: thesis:

end;

for x being Point of X
for a being Real holds f . (a * x) = a * (f . x)

proof

let x be Point of X; :: thesis:
let a be Real; :: thesis:
reconsider fx = f . x as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
reconsider fax = f . (a * x) as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
for y being Point of Y holds fax . y = a * (fx . y)

proof

let y be Point of Y; :: thesis:
A75: BL . (x,y) = fx . y by A60, A1;
BL . ((a * x),y) = fax . y by A60, A1;
hence fax . y = a * (fx . y) by A75, LOPBAN_8:12; :: thesis:

end;

hence f . (a * x) = a * (f . x) by LOPBAN_1:36; :: thesis:

end;

then reconsider f = f as LinearOperator of X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)) by A71, LOPBAN_1:def 5, VECTSP_1:def 20;
for x being Point of X holds ||.(f . x).|| <= ||.BLp.|| * ||.x.||

proof

let x be Point of X; :: thesis:
reconsider fx = f . x as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;
A78: for y being Point of Y holds ||.(fx . y).|| <= (||.BLp.|| * ||.x.||) * ||.y.||

proof

let y be Point of Y; :: thesis:
BL . (x,y) = fx . y by A60, A1;
hence ||.(fx . y).|| <= (||.BLp.|| * ||.x.||) * ||.y.|| by Th32; :: thesis:

end;

A79: ||.(f . x).|| = upper_bound (PreNorms fx) by LOPBAN_1:30;

now :: thesis:

let s be Real; :: thesis:
assume s in PreNorms fx ; :: thesis:
then consider y being Point of Y such that
A80: s = ||.(fx . y).|| and
A81: ||.y.|| <= 1 ;
A82: ||.(fx . y).|| <= (||.BLp.|| * ||.x.||) * ||.y.|| by A78;
(||.BLp.|| * ||.x.||) * ||.y.|| <= (||.BLp.|| * ||.x.||) * 1 by A81, XREAL_1:66;
hence s <= ||.BLp.|| * ||.x.|| by A80, A82, XXREAL_0:2; :: thesis:

end;

hence ||.(f . x).|| <= ||.BLp.|| * ||.x.|| by A79, SEQ_4:45; :: thesis:

end;

then reconsider f = f as Lipschitzian LinearOperator of X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)) by LOPBAN_1:def 8;
A83: f in the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))) by LOPBAN_1:def 9;
take f ; :: thesis:
thus ( f in the carrier of (R_NormSpace_of_BoundedLinearOperators (X,(R_NormSpace_of_BoundedLinearOperators (Y,Z)))) & y = I . f ) by A60, A83, FUNCT_1:49; :: thesis:

end;