let X be non empty set ; :: thesis: for Y being ComplexLinearSpace
for f, h being Element of Funcs (X, the carrier of Y)
for a being Complex holds
( h = (FuncExtMult (X,Y)) . [a,f] iff for x being Element of X holds h . x = a * (f . x) )

let Y be ComplexLinearSpace; :: thesis: for f, h being Element of Funcs (X, the carrier of Y)
for a being Complex holds
( h = (FuncExtMult (X,Y)) . [a,f] iff for x being Element of X holds h . x = a * (f . x) )

let f, h be Element of Funcs (X, the carrier of Y); :: thesis: for a being Complex holds
( h = (FuncExtMult (X,Y)) . [a,f] iff for x being Element of X holds h . x = a * (f . x) )

let a be Complex; :: thesis: ( h = (FuncExtMult (X,Y)) . [a,f] iff for x being Element of X holds h . x = a * (f . x) )
thus ( h = (FuncExtMult (X,Y)) . [a,f] implies for x being Element of X holds h . x = a * (f . x) ) by Def1; :: thesis: ( ( for x being Element of X holds h . x = a * (f . x) ) implies h = (FuncExtMult (X,Y)) . [a,f] )
reconsider a = a as Element of COMPLEX by XCMPLX_0:def 2;
now :: thesis: ( ( for x being Element of X holds h . x = a * (f . x) ) implies h = (FuncExtMult (X,Y)) . [a,f] )
assume A1: for x being Element of X holds h . x = a * (f . x) ; :: thesis: h = (FuncExtMult (X,Y)) . [a,f]
for x being Element of X holds h . x = ((FuncExtMult (X,Y)) . [a,f]) . x
proof
let x be Element of X; :: thesis: h . x = ((FuncExtMult (X,Y)) . [a,f]) . x
thus h . x = a * (f . x) by A1
.= ((FuncExtMult (X,Y)) . [a,f]) . x by Def1 ; :: thesis: verum
end;
hence h = (FuncExtMult (X,Y)) . [a,f] by FUNCT_2:63; :: thesis: verum
end;
hence ( ( for x being Element of X holds h . x = a * (f . x) ) implies h = (FuncExtMult (X,Y)) . [a,f] ) ; :: thesis: verum