reserve x,O for set,
  o for Element of O,
  G,H,I for GroupWithOperators of O,
  A, B for Subset of G,
  N for normal StableSubgroup of G,
  H1,H2,H3 for StableSubgroup of G,
  g1,g2 for Element of G,
  h1,h2 for Element of H1,
  h for Homomorphism of G,H;

theorem Th47:
  g1 in Ker h iff h.g1 = 1_H
proof
  thus g1 in Ker h implies h.g1 = 1_H
  proof
    assume g1 in Ker h;
    then g1 in the carrier of Ker h by STRUCT_0:def 5;
    then g1 in {b where b is Element of G : h.b = 1_H} by Def21;
    then ex b being Element of G st g1 = b & h.b = 1_H;
    hence thesis;
  end;
  assume h.g1 = 1_H;
  then g1 in {b where b is Element of G: h.b = 1_H};
  then g1 in the carrier of Ker h by Def21;
  hence thesis by STRUCT_0:def 5;
end;
