 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 Th19:
  for H being Subgroup of G
  holds incl (H, G) is one-to-one
  & Image (incl (H, G)) = the multMagma of H
proof
  let H be Subgroup of G;
  set f = incl (H, G);
  A1: f = id (the carrier of H) by Def9; then
  A2: the carrier of H = rng f
                      .= the carrier of (Image f) by GROUP_6:44;
  Ker f = (1).H
  proof
    for h being Element of H holds h in Ker f iff h in (1).H
    proof
      let h be Element of H;
      hereby
        assume h in Ker f;
        then f.h = 1_G by GROUP_6:41
                .= 1_H by GROUP_2:44;
        then h in {1_H} by A1, TARSKI:def 1;
        hence h in (1).H by GROUP_2:def 7;
      end;
      assume h in (1).H;
      then h in {1_H} by GROUP_2:def 7;
      then h = 1_H by TARSKI:def 1;
      then f.h = 1_G by GROUP_6:31;
      hence h in Ker f by GROUP_6:41;
    end;
    hence thesis by GROUP_2:def 6;
  end;
  hence f is one-to-one by GROUP_6:56;
  thus Image f = the multMagma of H by A2, GROUP_2:59;
end;
