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) ;
( h = c * f iff h1 = c * f1 )

proof

A1: now

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;

now

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;

hence ( h = c * f iff h1 = c * f1 ) by A1; :: thesis: verum

end;

hence ( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) ) by Th28; :: thesis: verum