theorem Th6:
  h1 = g1 implies h1" = g1"
proof
  reconsider g9 = h1" as Element of G by Th2;
A1: h1 * h1" = 1_H1 by GROUP_1:def 5;
  assume h1 = g1;
  then g1 * g9 = 1_H1 by A1,Th3
    .= 1_G by Th4;
  hence thesis by GROUP_1:12;
end;
