 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;

theorem
  for K being Group
  for H being Subgroup of G
  for phi being Homomorphism of G,K
  holds phi|H = phi * (incl H)
proof
  let K be Group;
  let H be Subgroup of G;
  let phi be Homomorphism of G,K;
  for h being Element of H holds
  (phi|H).h = (phi * (incl H)).h
  proof
    let h be Element of H;
    (incl H).h = (id (the carrier of H)).h
    & dom (incl H) = the carrier of H by Def9, FUNCT_2:def 1;
    then A1: (phi * (incl H)).h = phi.h by FUNCT_1:13;
    h in G by GROUP_2:41;
    then h in dom phi by FUNCT_2:def 1;
    then h in (dom phi) /\ (the carrier of H) by XBOOLE_0:def 4;
    then (phi|the carrier of H).h = phi.h by FUNCT_1:48;
    hence (phi|H).h = (phi * (incl H)).h by A1, GRSOLV_1:def 2;
  end;
  hence phi|H = phi * (incl H) by FUNCT_2:def 8;
end;
