 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem ThMorphismOfCommutators:
  for G1,G2 being Group
  for phi being Homomorphism of G1, G2
  for x being Element of G1 st x in commutators G1
  holds phi.x in commutators G2
proof
  let G1,G2 be Group;
  let phi be Homomorphism of G1, G2;
  let x be Element of G1;
  assume x in commutators G1;
  then consider a,b being Element of G1 such that
  A2: x = [. a, b .] by GROUP_5:58;
  phi.x = [. (phi.a), (phi.b) .] by A2, GROUP_6:34;
  hence phi.x in commutators G2 by GROUP_5:58;
end;
