let X, Y, Z be RealLinearSpace; :: thesis: ex I being LinearOperator of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))),(R_VectorSpace_of_BilinearOperators (X,Y,Z)) st

( I is bijective & ( for u being Point of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))))

for x being Point of X

for y being Point of Y holds (I . u) . (x,y) = (u . x) . y ) )

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_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))));

set BXYZ = the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z));

set LYZ = the carrier of (R_VectorSpace_of_LinearOperators (Y,Z));

then the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) c= Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))) by XBOOLE_1:1;

then reconsider I = I0 | the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) as Function of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))),(Funcs ([: the carrier of X, the carrier of Y:], the carrier of Z)) by FUNCT_2:32;

A7: for x being Element of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) holds

( ( for p being Point of X

for q being Point of Y ex G being LinearOperator of Y,Z st

( G = x . p & (I . x) . (p,q) = G . q ) ) & I . x in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) )

then reconsider I = I as Function of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))), the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by FUNCT_2:6;

A28: for x1, x2 being Element of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) holds I . (x1 + x2) = (I . x1) + (I . x2)

for a being Real holds I . (a * x) = a * (I . x)

A36: for u being Point of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))))

for x being Point of X

for y being Point of Y holds (I . u) . (x,y) = (u . x) . y

ex x being object st

( x in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) & y = I . x )

reconsider I = I as LinearOperator of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))),(R_VectorSpace_of_BilinearOperators (X,Y,Z)) ;

take I ; :: thesis: ( I is bijective & ( for u being Point of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))))

for x being Point of X

for y being Point of Y holds (I . u) . (x,y) = (u . x) . y ) )

( I is one-to-one & I is onto ) by A1, A58, FUNCT_1:52, FUNCT_2:def 3;

hence ( I is bijective & ( for u being Point of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))))

for x being Point of X

for y being Point of Y holds (I . u) . (x,y) = (u . x) . y ) ) by A36; :: thesis: verum

( I is bijective & ( for u being Point of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))))

for x being Point of X

for y being Point of Y holds (I . u) . (x,y) = (u . x) . y ) )

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_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))));

set BXYZ = the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z));

set LYZ = the carrier of (R_VectorSpace_of_LinearOperators (Y,Z));

now :: thesis: for x being object st x in Funcs ( the carrier of X, the carrier of (R_VectorSpace_of_LinearOperators (Y,Z))) holds

x in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)))

then A6:
Funcs ( the carrier of X, the carrier of (R_VectorSpace_of_LinearOperators (Y,Z))) c= Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)))
by TARSKI:def 3;x in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)))

let x be object ; :: thesis: ( x in Funcs ( the carrier of X, the carrier of (R_VectorSpace_of_LinearOperators (Y,Z))) implies x in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))) )

assume x in Funcs ( the carrier of X, the carrier of (R_VectorSpace_of_LinearOperators (Y,Z))) ; :: thesis: x in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)))

then consider f being Function such that

A5: ( x = f & dom f = the carrier of X & rng f c= the carrier of (R_VectorSpace_of_LinearOperators (Y,Z)) ) by FUNCT_2:def 2;

rng f c= Funcs ( the carrier of Y, the carrier of Z) by A5, XBOOLE_1:1;

hence x in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))) by A5, FUNCT_2:def 2; :: thesis: verum

end;assume x in Funcs ( the carrier of X, the carrier of (R_VectorSpace_of_LinearOperators (Y,Z))) ; :: thesis: x in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)))

then consider f being Function such that

A5: ( x = f & dom f = the carrier of X & rng f c= the carrier of (R_VectorSpace_of_LinearOperators (Y,Z)) ) by FUNCT_2:def 2;

rng f c= Funcs ( the carrier of Y, the carrier of Z) by A5, XBOOLE_1:1;

hence x in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))) by A5, FUNCT_2:def 2; :: thesis: verum

then the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) c= Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))) by XBOOLE_1:1;

then reconsider I = I0 | the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) as Function of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))),(Funcs ([: the carrier of X, the carrier of Y:], the carrier of Z)) by FUNCT_2:32;

A7: for x being Element of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) holds

( ( for p being Point of X

for q being Point of Y ex G being LinearOperator of Y,Z st

( G = x . p & (I . x) . (p,q) = G . q ) ) & I . x in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) )

proof

then
rng I c= the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z))
by FUNCT_2:114;
let f be Element of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))); :: thesis: ( ( for p being Point of X

for q being Point of Y ex G being LinearOperator of Y,Z st

( G = f . p & (I . f) . (p,q) = G . q ) ) & I . f in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) )

A8: I . f = I0 . f by FUNCT_1:49;

A9: f in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))) by A6, TARSKI:def 3, XBOOLE_1:1;

then A10: f is Function of the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)) by FUNCT_2:66;

reconsider g = f as Function of the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)) by A9, FUNCT_2:66;

reconsider F = f as LinearOperator of X,(R_VectorSpace_of_LinearOperators (Y,Z)) by LOPBAN_1:def 6;

thus for x being Point of X

for y being Point of Y ex G being LinearOperator of Y,Z st

( G = f . x & (I . f) . (x,y) = G . y ) :: thesis: I . f in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z))

A14: for x1, x2 being Point of X

for y being Point of Y holds BL . ((x1 + x2),y) = (BL . (x1,y)) + (BL . (x2,y))

for y being Point of Y

for a being Real holds BL . ((a * x),y) = a * (BL . (x,y))

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

for y being Point of Y

for a being Real holds BL . (x,(a * y)) = a * (BL . (x,y))

BL in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by Def6;

hence I . f in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by FUNCT_1:49; :: thesis: verum

end;for q being Point of Y ex G being LinearOperator of Y,Z st

( G = f . p & (I . f) . (p,q) = G . q ) ) & I . f in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) )

A8: I . f = I0 . f by FUNCT_1:49;

A9: f in Funcs ( the carrier of X,(Funcs ( the carrier of Y, the carrier of Z))) by A6, TARSKI:def 3, XBOOLE_1:1;

then A10: f is Function of the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)) by FUNCT_2:66;

reconsider g = f as Function of the carrier of X,(Funcs ( the carrier of Y, the carrier of Z)) by A9, FUNCT_2:66;

reconsider F = f as LinearOperator of X,(R_VectorSpace_of_LinearOperators (Y,Z)) by LOPBAN_1:def 6;

thus for x being Point of X

for y being Point of Y ex G being LinearOperator of Y,Z st

( G = f . x & (I . f) . (x,y) = G . y ) :: thesis: I . f in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z))

proof

reconsider BL = I0 . f as Function of [:X,Y:],Z by A9, FUNCT_2:5, FUNCT_2:66;
let x be Point of X; :: thesis: for y being Point of Y ex G being LinearOperator of Y,Z st

( G = f . x & (I . f) . (x,y) = G . y )

let y be Point of Y; :: thesis: ex G being LinearOperator of Y,Z st

( G = f . x & (I . f) . (x,y) = G . y )

g . x = F . x ;

then reconsider G = g . x as LinearOperator of Y,Z by LOPBAN_1:def 6;

take G ; :: thesis: ( G = f . x & (I . f) . (x,y) = G . y )

thus ( G = f . x & (I . f) . (x,y) = G . y ) by A1, A8; :: thesis: verum

end;( G = f . x & (I . f) . (x,y) = G . y )

let y be Point of Y; :: thesis: ex G being LinearOperator of Y,Z st

( G = f . x & (I . f) . (x,y) = G . y )

g . x = F . x ;

then reconsider G = g . x as LinearOperator of Y,Z by LOPBAN_1:def 6;

take G ; :: thesis: ( G = f . x & (I . f) . (x,y) = G . y )

thus ( G = f . x & (I . f) . (x,y) = G . y ) by A1, A8; :: thesis: verum

A14: 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

A18:
for x being Point of X
let x1, x2 be Point of X; :: thesis: for y being Point of Y holds BL . ((x1 + x2),y) = (BL . (x1,y)) + (BL . (x2,y))

let y be Point of Y; :: thesis: BL . ((x1 + x2),y) = (BL . (x1,y)) + (BL . (x2,y))

A15: BL . (x1,y) = (F . x1) . y by A1, A10;

A16: BL . (x2,y) = (F . x2) . y by A1, A10;

A17: BL . ((x1 + x2),y) = (F . (x1 + x2)) . y by A1, A10;

F . (x1 + x2) = (F . x1) + (F . x2) by VECTSP_1:def 20;

hence BL . ((x1 + x2),y) = (BL . (x1,y)) + (BL . (x2,y)) by A15, A16, A17, LOPBAN_1:16; :: thesis: verum

end;let y be Point of Y; :: thesis: BL . ((x1 + x2),y) = (BL . (x1,y)) + (BL . (x2,y))

A15: BL . (x1,y) = (F . x1) . y by A1, A10;

A16: BL . (x2,y) = (F . x2) . y by A1, A10;

A17: BL . ((x1 + x2),y) = (F . (x1 + x2)) . y by A1, A10;

F . (x1 + x2) = (F . x1) + (F . x2) by VECTSP_1:def 20;

hence BL . ((x1 + x2),y) = (BL . (x1,y)) + (BL . (x2,y)) by A15, A16, A17, LOPBAN_1:16; :: thesis: verum

for y being Point of Y

for a being Real holds BL . ((a * x),y) = a * (BL . (x,y))

proof

A21:
for x being Point of X
let x be Point of X; :: thesis: for y being Point of Y

for a being Real holds BL . ((a * x),y) = a * (BL . (x,y))

let y be Point of Y; :: thesis: for a being Real holds BL . ((a * x),y) = a * (BL . (x,y))

let a be Real; :: thesis: BL . ((a * x),y) = a * (BL . (x,y))

A19: BL . ((a * x),y) = (F . (a * x)) . y by A1, A10;

A20: BL . (x,y) = (F . x) . y by A1, A10;

F . (a * x) = a * (F . x) by LOPBAN_1:def 5;

hence BL . ((a * x),y) = a * (BL . (x,y)) by A19, A20, LOPBAN_1:17; :: thesis: verum

end;for a being Real holds BL . ((a * x),y) = a * (BL . (x,y))

let y be Point of Y; :: thesis: for a being Real holds BL . ((a * x),y) = a * (BL . (x,y))

let a be Real; :: thesis: BL . ((a * x),y) = a * (BL . (x,y))

A19: BL . ((a * x),y) = (F . (a * x)) . y by A1, A10;

A20: BL . (x,y) = (F . x) . y by A1, A10;

F . (a * x) = a * (F . x) by LOPBAN_1:def 5;

hence BL . ((a * x),y) = a * (BL . (x,y)) by A19, A20, LOPBAN_1:17; :: thesis: verum

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

proof

A25:
for x being Point of X
let x be Point of X; :: thesis: for y1, y2 being Point of Y holds BL . (x,(y1 + y2)) = (BL . (x,y1)) + (BL . (x,y2))

let y1, y2 be Point of Y; :: thesis: BL . (x,(y1 + y2)) = (BL . (x,y1)) + (BL . (x,y2))

reconsider Fx = F . x as LinearOperator of Y,Z by LOPBAN_1:def 6;

A22: BL . (x,y1) = Fx . y1 by A1, A10;

A23: BL . (x,y2) = Fx . y2 by A1, A10;

BL . (x,(y1 + y2)) = Fx . (y1 + y2) by A1, A10;

hence BL . (x,(y1 + y2)) = (BL . (x,y1)) + (BL . (x,y2)) by A22, A23, VECTSP_1:def 20; :: thesis: verum

end;let y1, y2 be Point of Y; :: thesis: BL . (x,(y1 + y2)) = (BL . (x,y1)) + (BL . (x,y2))

reconsider Fx = F . x as LinearOperator of Y,Z by LOPBAN_1:def 6;

A22: BL . (x,y1) = Fx . y1 by A1, A10;

A23: BL . (x,y2) = Fx . y2 by A1, A10;

BL . (x,(y1 + y2)) = Fx . (y1 + y2) by A1, A10;

hence BL . (x,(y1 + y2)) = (BL . (x,y1)) + (BL . (x,y2)) by A22, A23, VECTSP_1:def 20; :: thesis: verum

for y being Point of Y

for a being Real holds BL . (x,(a * y)) = a * (BL . (x,y))

proof

reconsider BL = BL as BilinearOperator of X,Y,Z by A14, A18, A21, A25, LOPBAN_8:11;
let x be Point of X; :: thesis: for y being Point of Y

for a being Real holds BL . (x,(a * y)) = a * (BL . (x,y))

let y be Point of Y; :: thesis: for a being Real holds BL . (x,(a * y)) = a * (BL . (x,y))

let a be Real; :: thesis: BL . (x,(a * y)) = a * (BL . (x,y))

reconsider Fx = F . x as LinearOperator of Y,Z by LOPBAN_1:def 6;

A26: BL . (x,y) = Fx . y by A1, A10;

BL . (x,(a * y)) = Fx . (a * y) by A1, A10;

hence BL . (x,(a * y)) = a * (BL . (x,y)) by A26, LOPBAN_1:def 5; :: thesis: verum

end;for a being Real holds BL . (x,(a * y)) = a * (BL . (x,y))

let y be Point of Y; :: thesis: for a being Real holds BL . (x,(a * y)) = a * (BL . (x,y))

let a be Real; :: thesis: BL . (x,(a * y)) = a * (BL . (x,y))

reconsider Fx = F . x as LinearOperator of Y,Z by LOPBAN_1:def 6;

A26: BL . (x,y) = Fx . y by A1, A10;

BL . (x,(a * y)) = Fx . (a * y) by A1, A10;

hence BL . (x,(a * y)) = a * (BL . (x,y)) by A26, LOPBAN_1:def 5; :: thesis: verum

BL in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by Def6;

hence I . f in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by FUNCT_1:49; :: thesis: verum

then reconsider I = I as Function of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))), the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by FUNCT_2:6;

A28: for x1, x2 being Element of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) holds I . (x1 + x2) = (I . x1) + (I . x2)

proof

for x being Element of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))))
let x1, x2 be Element of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))); :: thesis: I . (x1 + x2) = (I . x1) + (I . x2)

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))

end;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

hence
I . (x1 + x2) = (I . x1) + (I . x2)
by Th16; :: thesis: verum
let p be Point of X; :: thesis: for q being Point of Y holds (I . (x1 + x2)) . (p,q) = ((I . x1) . (p,q)) + ((I . x2) . (p,q))

let q be Point of Y; :: thesis: (I . (x1 + x2)) . (p,q) = ((I . x1) . (p,q)) + ((I . x2) . (p,q))

consider Gx1p being LinearOperator of Y,Z such that

A29: ( Gx1p = x1 . p & (I . x1) . (p,q) = Gx1p . q ) by A7;

consider Gx2p being LinearOperator of Y,Z such that

A30: ( Gx2p = x2 . p & (I . x2) . (p,q) = Gx2p . q ) by A7;

consider Gx1x2p being LinearOperator of Y,Z such that

A31: ( Gx1x2p = (x1 + x2) . p & (I . (x1 + x2)) . (p,q) = Gx1x2p . q ) by A7;

(x1 + x2) . p = (x1 . p) + (x2 . p) by LOPBAN_1:16;

hence (I . (x1 + x2)) . (p,q) = ((I . x1) . (p,q)) + ((I . x2) . (p,q)) by A29, A30, A31, LOPBAN_1:16; :: thesis: verum

end;let q be Point of Y; :: thesis: (I . (x1 + x2)) . (p,q) = ((I . x1) . (p,q)) + ((I . x2) . (p,q))

consider Gx1p being LinearOperator of Y,Z such that

A29: ( Gx1p = x1 . p & (I . x1) . (p,q) = Gx1p . q ) by A7;

consider Gx2p being LinearOperator of Y,Z such that

A30: ( Gx2p = x2 . p & (I . x2) . (p,q) = Gx2p . q ) by A7;

consider Gx1x2p being LinearOperator of Y,Z such that

A31: ( Gx1x2p = (x1 + x2) . p & (I . (x1 + x2)) . (p,q) = Gx1x2p . q ) by A7;

(x1 + x2) . p = (x1 . p) + (x2 . p) by LOPBAN_1:16;

hence (I . (x1 + x2)) . (p,q) = ((I . x1) . (p,q)) + ((I . x2) . (p,q)) by A29, A30, A31, LOPBAN_1:16; :: thesis: verum

for a being Real holds I . (a * x) = a * (I . x)

proof

then reconsider I = I as LinearOperator of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))),(R_VectorSpace_of_BilinearOperators (X,Y,Z)) by A28, LOPBAN_1:def 5, VECTSP_1:def 20;
let x be Element of the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))); :: thesis: for a being Real holds I . (a * x) = a * (I . x)

let a be Real; :: thesis: I . (a * x) = a * (I . x)

for p being Point of X

for q being Point of Y holds (I . (a * x)) . (p,q) = a * ((I . x) . (p,q))

end;let a be Real; :: thesis: I . (a * x) = a * (I . x)

for p being Point of X

for q being Point of Y holds (I . (a * x)) . (p,q) = a * ((I . x) . (p,q))

proof

hence
I . (a * x) = a * (I . x)
by Th17; :: thesis: verum
let p be Point of X; :: thesis: for q being Point of Y holds (I . (a * x)) . (p,q) = a * ((I . x) . (p,q))

let q be Point of Y; :: thesis: (I . (a * x)) . (p,q) = a * ((I . x) . (p,q))

consider Gxp being LinearOperator of Y,Z such that

A33: ( Gxp = x . p & (I . x) . (p,q) = Gxp . q ) by A7;

consider Gxap being LinearOperator of Y,Z such that

A34: ( Gxap = (a * x) . p & (I . (a * x)) . (p,q) = Gxap . q ) by A7;

(a * x) . p = a * (x . p) by LOPBAN_1:17;

hence (I . (a * x)) . (p,q) = a * ((I . x) . (p,q)) by A33, A34, LOPBAN_1:17; :: thesis: verum

end;let q be Point of Y; :: thesis: (I . (a * x)) . (p,q) = a * ((I . x) . (p,q))

consider Gxp being LinearOperator of Y,Z such that

A33: ( Gxp = x . p & (I . x) . (p,q) = Gxp . q ) by A7;

consider Gxap being LinearOperator of Y,Z such that

A34: ( Gxap = (a * x) . p & (I . (a * x)) . (p,q) = Gxap . q ) by A7;

(a * x) . p = a * (x . p) by LOPBAN_1:17;

hence (I . (a * x)) . (p,q) = a * ((I . x) . (p,q)) by A33, A34, LOPBAN_1:17; :: thesis: verum

A36: for u being Point of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))))

for x being Point of X

for y being Point of Y holds (I . u) . (x,y) = (u . x) . y

proof

for y being object st y in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) holds
let u be Point of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))); :: thesis: for x being Point of X

for y being Point of Y holds (I . u) . (x,y) = (u . x) . y

let p be Point of X; :: thesis: for y being Point of Y holds (I . u) . (p,y) = (u . p) . y

let q be Point of Y; :: thesis: (I . u) . (p,q) = (u . p) . q

consider G being LinearOperator of Y,Z such that

A37: ( G = u . p & (I . u) . (p,q) = G . q ) by A7;

thus (I . u) . (p,q) = (u . p) . q by A37; :: thesis: verum

end;for y being Point of Y holds (I . u) . (x,y) = (u . x) . y

let p be Point of X; :: thesis: for y being Point of Y holds (I . u) . (p,y) = (u . p) . y

let q be Point of Y; :: thesis: (I . u) . (p,q) = (u . p) . q

consider G being LinearOperator of Y,Z such that

A37: ( G = u . p & (I . u) . (p,q) = G . q ) by A7;

thus (I . u) . (p,q) = (u . p) . q by A37; :: thesis: verum

ex x being object st

( x in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) & y = I . x )

proof

then A58:
rng I = the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z))
by FUNCT_2:10;
let y be object ; :: thesis: ( y in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) implies ex x being object st

( x in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) & y = I . x ) )

assume A39: y in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) ; :: thesis: ex x being object st

( x in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) & y = I . x )

then y in Funcs ([: the carrier of X, the carrier of Y:], the carrier of Z) ;

then y in rng I0 by A1, FUNCT_2:def 3;

then consider f being object such that

A40: ( 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 A40, FUNCT_2:66;

reconsider BL = y as BilinearOperator of X,Y,Z by A39, Def6;

reconsider BLp = BL as Point of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by Def6;

A42: 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_VectorSpace_of_LinearOperators (Y,Z))

A50: for x1, x2 being Point of X holds f . (x1 + x2) = (f . x1) + (f . x2)

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

A56: f in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) by LOPBAN_1:def 6;

take f ; :: thesis: ( f in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) & y = I . f )

thus ( f in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) & y = I . f ) by A40, A56, FUNCT_1:49; :: thesis: verum

end;( x in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) & y = I . x ) )

assume A39: y in the carrier of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) ; :: thesis: ex x being object st

( x in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) & y = I . x )

then y in Funcs ([: the carrier of X, the carrier of Y:], the carrier of Z) ;

then y in rng I0 by A1, FUNCT_2:def 3;

then consider f being object such that

A40: ( 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 A40, FUNCT_2:66;

reconsider BL = y as BilinearOperator of X,Y,Z by A39, Def6;

reconsider BLp = BL as Point of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by Def6;

A42: 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_VectorSpace_of_LinearOperators (Y,Z))

proof

then reconsider f = f as Function of the carrier of X, the carrier of (R_VectorSpace_of_LinearOperators (Y,Z)) by A42, FUNCT_2:3;
let x be object ; :: thesis: ( x in the carrier of X implies f . x in the carrier of (R_VectorSpace_of_LinearOperators (Y,Z)) )

assume A43: x in the carrier of X ; :: thesis: f . x in the carrier of (R_VectorSpace_of_LinearOperators (Y,Z))

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 A43;

A44: for p, q being Point of Y holds fx . (p + q) = (fx . p) + (fx . q)

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

fx in the carrier of (R_VectorSpace_of_LinearOperators (Y,Z)) by LOPBAN_1:def 6;

hence f . x in the carrier of (R_VectorSpace_of_LinearOperators (Y,Z)) ; :: thesis: verum

end;assume A43: x in the carrier of X ; :: thesis: f . x in the carrier of (R_VectorSpace_of_LinearOperators (Y,Z))

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 A43;

A44: for p, q being Point of Y holds fx . (p + q) = (fx . p) + (fx . q)

proof

for p being Point of Y
let p, q be Point of Y; :: thesis: fx . (p + q) = (fx . p) + (fx . q)

A45: BL . (xp,p) = fx . p by A1, A40;

A46: BL . (xp,q) = fx . q by A1, A40;

BL . (xp,(p + q)) = fx . (p + q) by A1, A40;

hence fx . (p + q) = (fx . p) + (fx . q) by A45, A46, LOPBAN_8:11; :: thesis: verum

end;A45: BL . (xp,p) = fx . p by A1, A40;

A46: BL . (xp,q) = fx . q by A1, A40;

BL . (xp,(p + q)) = fx . (p + q) by A1, A40;

hence fx . (p + q) = (fx . p) + (fx . q) by A45, A46, LOPBAN_8:11; :: thesis: verum

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

proof

then reconsider fx = fx as LinearOperator of Y,Z by A44, LOPBAN_1:def 5, VECTSP_1:def 20;
let p be Point of Y; :: thesis: for a being Real holds fx . (a * p) = a * (fx . p)

let a be Real; :: thesis: fx . (a * p) = a * (fx . p)

A48: BL . (xp,p) = fx . p by A1, A40;

BL . (xp,(a * p)) = fx . (a * p) by A1, A40;

hence fx . (a * p) = a * (fx . p) by A48, LOPBAN_8:11; :: thesis: verum

end;let a be Real; :: thesis: fx . (a * p) = a * (fx . p)

A48: BL . (xp,p) = fx . p by A1, A40;

BL . (xp,(a * p)) = fx . (a * p) by A1, A40;

hence fx . (a * p) = a * (fx . p) by A48, LOPBAN_8:11; :: thesis: verum

fx in the carrier of (R_VectorSpace_of_LinearOperators (Y,Z)) by LOPBAN_1:def 6;

hence f . x in the carrier of (R_VectorSpace_of_LinearOperators (Y,Z)) ; :: thesis: verum

A50: for x1, x2 being Point of X holds f . (x1 + x2) = (f . x1) + (f . x2)

proof

for x being Point of X
let x1, x2 be Point of X; :: thesis: f . (x1 + x2) = (f . x1) + (f . x2)

reconsider fx1x2 = f . (x1 + x2) as LinearOperator of Y,Z by LOPBAN_1:def 6;

reconsider fx1 = f . x1 as LinearOperator of Y,Z by LOPBAN_1:def 6;

reconsider fx2 = f . x2 as LinearOperator of Y,Z by LOPBAN_1:def 6;

for y being Point of Y holds fx1x2 . y = (fx1 . y) + (fx2 . y)

end;reconsider fx1x2 = f . (x1 + x2) as LinearOperator of Y,Z by LOPBAN_1:def 6;

reconsider fx1 = f . x1 as LinearOperator of Y,Z by LOPBAN_1:def 6;

reconsider fx2 = f . x2 as LinearOperator of Y,Z by LOPBAN_1:def 6;

for y being Point of Y holds fx1x2 . y = (fx1 . y) + (fx2 . y)

proof

hence
f . (x1 + x2) = (f . x1) + (f . x2)
by LOPBAN_1:16; :: thesis: verum
let y be Point of Y; :: thesis: fx1x2 . y = (fx1 . y) + (fx2 . y)

A51: BL . (x1,y) = fx1 . y by A1, A40;

A52: BL . (x2,y) = fx2 . y by A1, A40;

BL . ((x1 + x2),y) = fx1x2 . y by A1, A40;

hence fx1x2 . y = (fx1 . y) + (fx2 . y) by A51, A52, LOPBAN_8:11; :: thesis: verum

end;A51: BL . (x1,y) = fx1 . y by A1, A40;

A52: BL . (x2,y) = fx2 . y by A1, A40;

BL . ((x1 + x2),y) = fx1x2 . y by A1, A40;

hence fx1x2 . y = (fx1 . y) + (fx2 . y) by A51, A52, LOPBAN_8:11; :: thesis: verum

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

proof

then reconsider f = f as LinearOperator of X,(R_VectorSpace_of_LinearOperators (Y,Z)) by A50, LOPBAN_1:def 5, VECTSP_1:def 20;
let x be Point of X; :: thesis: for a being Real holds f . (a * x) = a * (f . x)

let a be Real; :: thesis: f . (a * x) = a * (f . x)

reconsider fx = f . x as LinearOperator of Y,Z by LOPBAN_1:def 6;

reconsider fax = f . (a * x) as LinearOperator of Y,Z by LOPBAN_1:def 6;

for y being Point of Y holds fax . y = a * (fx . y)

end;let a be Real; :: thesis: f . (a * x) = a * (f . x)

reconsider fx = f . x as LinearOperator of Y,Z by LOPBAN_1:def 6;

reconsider fax = f . (a * x) as LinearOperator of Y,Z by LOPBAN_1:def 6;

for y being Point of Y holds fax . y = a * (fx . y)

proof

hence
f . (a * x) = a * (f . x)
by LOPBAN_1:17; :: thesis: verum
let y be Point of Y; :: thesis: fax . y = a * (fx . y)

A54: BL . (x,y) = fx . y by A1, A40;

BL . ((a * x),y) = fax . y by A1, A40;

hence fax . y = a * (fx . y) by A54, LOPBAN_8:11; :: thesis: verum

end;A54: BL . (x,y) = fx . y by A1, A40;

BL . ((a * x),y) = fax . y by A1, A40;

hence fax . y = a * (fx . y) by A54, LOPBAN_8:11; :: thesis: verum

A56: f in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) by LOPBAN_1:def 6;

take f ; :: thesis: ( f in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) & y = I . f )

thus ( f in the carrier of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))) & y = I . f ) by A40, A56, FUNCT_1:49; :: thesis: verum

reconsider I = I as LinearOperator of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z)))),(R_VectorSpace_of_BilinearOperators (X,Y,Z)) ;

take I ; :: thesis: ( I is bijective & ( for u being Point of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))))

for x being Point of X

for y being Point of Y holds (I . u) . (x,y) = (u . x) . y ) )

( I is one-to-one & I is onto ) by A1, A58, FUNCT_1:52, FUNCT_2:def 3;

hence ( I is bijective & ( for u being Point of (R_VectorSpace_of_LinearOperators (X,(R_VectorSpace_of_LinearOperators (Y,Z))))

for x being Point of X

for y being Point of Y holds (I . u) . (x,y) = (u . x) . y ) ) by A36; :: thesis: verum