theorem Th48:
  h = g implies h" = g"
proof
  reconsider g9 = h" as Element of G by Th42;
A1: h * h" = 1_H by GROUP_1:def 5;
  assume h = g;
  then g * g9 = 1_H by A1,Th43
    .= 1_G by Th44;
  hence thesis by GROUP_1:12;
end;
