let X be non empty set ; :: thesis: for Y being ComplexLinearSpace
for f, h being VECTOR of (ComplexVectSpace X,Y)
for c being Complex holds
( h = c * f iff for x being Element of X holds h . x = c * (f . x) )
let Y be ComplexLinearSpace; :: thesis: for f, h being VECTOR of (ComplexVectSpace X,Y)
for c being Complex holds
( h = c * f iff for x being Element of X holds h . x = c * (f . x) )
let f, h be VECTOR of (ComplexVectSpace X,Y); :: thesis: for c being Complex holds
( h = c * f iff for x being Element of X holds h . x = c * (f . x) )
let c be Complex; :: thesis: ( h = c * f iff for x being Element of X holds h . x = c * (f . x) )
reconsider f' = f, h' = h as Element of Funcs X,the carrier of Y ;

hereby :: thesis: ( ( for x being Element of X holds h . x = c * (f . x) ) implies h = c * f )

assume A1: h = c * f ; :: thesis: for x being Element of X holds h . x = c * (f . x)
let x be Element of X; :: thesis: h . x = c * (f . x)
h = (FuncExtMult X,Y) . [c,f'] by A1, CLVECT_1:def 1;
hence h . x = c * (f . x) by Th3; :: thesis: verum

end;

assume for x being Element of X holds h . x = c * (f . x) ; :: thesis: h = c * f
then h' = (FuncExtMult X,Y) . [c,f'] by Th3;
hence h = c * f by CLVECT_1:def 1; :: thesis: verum