let IT1, IT2 be Function; :: thesis: ( dom IT1 = dom f & ( for x being set st x in dom f holds
IT1 . x = g * (f . x) ) & dom IT2 = dom f & ( for x being set st x in dom f holds
IT2 . x = g * (f . x) ) implies IT1 = IT2 )
assume B1: ( dom IT1 = dom f & ( for x being set st x in dom f holds
IT1 . x = g * (f . x) ) ) ; :: thesis: ( not dom IT2 = dom f or ex x being set st
( x in dom f & not IT2 . x = g * (f . x) ) or IT1 = IT2 )
assume B2: ( dom IT2 = dom f & ( for x being set st x in dom f holds
IT2 . x = g * (f . x) ) ) ; :: thesis: IT1 = IT2

now

let x be set ; :: thesis: ( x in dom IT1 implies IT1 . x = IT2 . x )
assume x in dom IT1 ; :: thesis: IT1 . x = IT2 . x
then ( IT1 . x = g * (f . x) & IT2 . x = g * (f . x) ) by B1, B2;
hence IT1 . x = IT2 . x ; :: thesis: verum

end;

hence IT1 = IT2 by B1, B2, FUNCT_1:2; :: thesis: verum