 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 H,K being Group st H is Subgroup of G
  for phi being Homomorphism of G,K
  holds phi|(the carrier of H) = phi * (incl (H, G))
proof
  let H,K be Group;
  assume A1: H is Subgroup of G;
  let phi be Homomorphism of G,K;
  A2: dom (phi|(the carrier of H)) = the carrier of H
  proof
    dom phi = the carrier of G by FUNCT_2:def 1;
    hence dom (phi|(the carrier of H)) = the carrier of H
    by A1, GROUP_2:def 5, RELAT_1:62;
  end;
  for x being object st x in dom (phi|(the carrier of H))
  holds (phi|(the carrier of H)).x = (phi * (incl (H, G))).x
  proof
    let x be object;
    assume B1: x in dom (phi|(the carrier of H));
    hence (phi * (incl (H, G))).x
     = phi.((incl (H, G)).x) by A2, FUNCT_2:15
    .= phi.x by A1, A2, B1, Th18
    .= (phi|(the carrier of H)).x by B1, FUNCT_1:47;
  end;
  hence phi|(the carrier of H) = phi * (incl (H, G))
  by A2, FUNCT_2:def 1;
end;
