theorem Th79:
  a * carr (Omega).G = {b : a,b are_conjugated}
proof
  set A = a * carr (Omega).G;
  set B = {b : a,b are_conjugated};
  thus A c= B
  proof
    let x be object;
    assume
A1: x in A;
    then reconsider b = x as Element of G;
    ex g st x = a * g & g in carr(Omega).G by A1,ThB42;
    then b,a are_conjugated;
    hence thesis;
  end;
  let x be object;
  assume x in B;
  then consider b such that
A2: x = b and
A3: a,b are_conjugated;
  ex g st b = a * g by A3,Def7;
  hence thesis by A2,ThB42;
end;
