theorem Th41:
  a in Ker h iff h.a = 1_H
proof
  thus a in Ker h implies h.a = 1_H
  proof
    assume a in Ker h;
    then a in the carrier of Ker h;
    then a in {b : h.b = 1_H} by Def9;
    then ex b st a = b & h.b = 1_H;
    hence thesis;
  end;
  assume h.a = 1_H;
  then a in {b : h.b = 1_H};
  then a in the carrier of Ker h by Def9;
  hence thesis;
end;
