theorem
  a<>0.G implies a"*a=1.G & a*(a") = 1.G
proof
  assume
A1: a<>0.G;
  hence
A2: a"*a = 1.G by Def10;
  consider x such that
A3: a*x = 1.G by A1,Def8;
  a"*a*x = a" * 1.G by A3,GROUP_1:def 3;
  then x = a" * 1.G by A2;
  hence thesis by A3;
end;
