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;
