let A be non empty set ; :: thesis: for f, h being Element of Funcs (A,COMPLEX)
for a being Complex holds
( h = . [a,f] iff for x being Element of A holds h . x = a * (f . x) )

let f, h be Element of Funcs (A,COMPLEX); :: thesis: for a being Complex holds
( h = . [a,f] iff for x being Element of A holds h . x = a * (f . x) )

let a be Complex; :: thesis: ( h = . [a,f] iff for x being Element of A holds h . x = a * (f . x) )
thus ( h = . [a,f] implies for x being Element of A holds h . x = a * (f . x) ) by Def3; :: thesis: ( ( for x being Element of A holds h . x = a * (f . x) ) implies h = . [a,f] )
reconsider a = a as Element of COMPLEX by XCMPLX_0:def 2;
now :: thesis: ( ( for x being Element of A holds h . x = a * (f . x) ) implies h = . [a,f] )
assume A1: for x being Element of A holds h . x = a * (f . x) ; :: thesis: h = . [a,f]
for x being Element of A holds h . x = ( . [a,f]) . x
proof
let x be Element of A; :: thesis: h . x = ( . [a,f]) . x
thus h . x = a * (f . x) by A1
.= ( . [a,f]) . x by Def3 ; :: thesis: verum
end;
hence h = . [a,f] by FUNCT_2:63; :: thesis: verum
end;
hence ( ( for x being Element of A holds h . x = a * (f . x) ) implies h = . [a,f] ) ; :: thesis: verum