let X, Y be ComplexNormSpace; :: thesis: for f, g, h being VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)) holds

( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

let f, g, h be VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)); :: thesis: ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

A1: C_VectorSpace_of_BoundedLinearOperators (X,Y) is Subspace of C_VectorSpace_of_LinearOperators (X,Y) by Th21, CSSPACE:11;

then reconsider f1 = f as VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) by CLVECT_1:29;

reconsider h1 = h as VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) by A1, CLVECT_1:29;

reconsider g1 = g as VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) by A1, CLVECT_1:29;

then h1 = f1 + g1 by Th15;

hence h = f + g by A1, CLVECT_1:32; :: thesis: verum

( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

let f, g, h be VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)); :: thesis: ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

A1: C_VectorSpace_of_BoundedLinearOperators (X,Y) is Subspace of C_VectorSpace_of_LinearOperators (X,Y) by Th21, CSSPACE:11;

then reconsider f1 = f as VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) by CLVECT_1:29;

reconsider h1 = h as VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) by A1, CLVECT_1:29;

reconsider g1 = g as VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) by A1, CLVECT_1:29;

hereby :: thesis: ( ( for x being VECTOR of X holds h . x = (f . x) + (g . x) ) implies h = f + g )

assume
for x being Element of X holds h . x = (f . x) + (g . x)
; :: thesis: h = f + gassume A2:
h = f + g
; :: thesis: for x being Element of X holds h . x = (f . x) + (g . x)

let x be Element of X; :: thesis: h . x = (f . x) + (g . x)

h1 = f1 + g1 by A1, A2, CLVECT_1:32;

hence h . x = (f . x) + (g . x) by Th15; :: thesis: verum

end;let x be Element of X; :: thesis: h . x = (f . x) + (g . x)

h1 = f1 + g1 by A1, A2, CLVECT_1:32;

hence h . x = (f . x) + (g . x) by Th15; :: thesis: verum

then h1 = f1 + g1 by Th15;

hence h = f + g by A1, CLVECT_1:32; :: thesis: verum