let f1, f2 be Function of X,L; :: thesis: ( ( for x being Element of X holds f1 . x = a * (f . x) ) & ( for x being Element of X holds f2 . x = a * (f . x) ) implies f1 = f2 )
assume A: for x being Element of X holds f1 . x = a * (f . x) ; :: thesis: ( ex x being Element of X st not f2 . x = a * (f . x) or f1 = f2 )
assume B: for x being Element of X holds f2 . x = a * (f . x) ; :: thesis: f1 = f2

now :: thesis: for o being object st o in X holds
f1 . o = f2 . o

let o be object ; :: thesis: ( o in X implies f1 . o = f2 . o )
assume o in X ; :: thesis: f1 . o = f2 . o
then reconsider x = o as Element of X ;
thus f1 . o = a * (f . x) by A
.= f2 . o by B ; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:12; :: thesis: verum