let X, Y be ComplexNormSpace; :: thesis: for f, h being Point of (C_NormSpace_of_BoundedLinearOperators (X,Y))
for c being Complex holds
( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
let f, h be Point of (C_NormSpace_of_BoundedLinearOperators (X,Y)); :: thesis: for c being Complex holds
( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
let c be Complex; :: thesis: ( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
reconsider f1 = f as VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)) ;
reconsider h1 = h as VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)) ;

A1: now :: thesis: ( h1 = c * f1 implies h = c * f )

assume h1 = c * f1 ; :: thesis: h = c * f
hence h = (Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) . [c,f1] by CLVECT_1:def 1
.= c * f by CLVECT_1:def 1 ;
:: thesis: verum

end;

now :: thesis: ( h = c * f implies h1 = c * f1 )

assume h = c * f ; :: thesis: h1 = c * f1
hence h1 = (Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) . [c,f] by CLVECT_1:def 1
.= c * f1 by CLVECT_1:def 1 ;
:: thesis: verum

end;

hence ( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) ) by A1, Th24; :: thesis: verum