theorem Th13:
  f * h = g iff f = g * h"
proof
  g * h" * h = g * (h" * h) by Def3
    .= g * 1_G by Def5
    .= g by Def4;
  hence f * h = g implies f = g * h" by Th6;
  assume f = g * h";
  hence f * h = g * (h" * h) by Def3
    .= g * 1_G by Def5
    .= g by Def4;
end;
