let X be non empty set ; :: thesis: for Y being ComplexLinearSpace
for h, f 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 h, f 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 h, f 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] )

now

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_2:113; :: 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