let F, G be Function of V1,V2; :: thesis: ( ( for v being Element of V1 holds F . v = a * (f . v) ) & ( for v being Element of V1 holds G . v = a * (f . v) ) implies F = G )
assume that
A2: for v being Element of V1 holds F . v = a * (f . v) and
A3: for v being Element of V1 holds G . v = a * (f . v) ; :: thesis: F = G

now

let v be Element of V1; :: thesis: F . v = G . v
thus F . v = a * (f . v) by A2
.= G . v by A3 ; :: thesis: verum

end;

hence F = G by FUNCT_2:113; :: thesis: verum