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 f9 = f, h9 = 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,f9] by A1, CLVECT_1:def 1;
hence h . x = c * (f . x) by Th2; :: thesis: verum

end;

assume for x being Element of X holds h . x = c * (f . x) ; :: thesis: h = c * f
then h9 = (FuncExtMult (X,Y)) . [c,f9] by Th2;
hence h = c * f by CLVECT_1:def 1; :: thesis: verum