theorem Th27:
  p ## q implies p~ = q~
proof
  assume
A1: p ## q;
  for x being object holds x in p~ iff x in q~
  proof
    let x be object;
    thus x in p~ implies x in q~
    proof
      assume
A2:   x in p~;
      then reconsider
      r = x as Element of [:the carrier of M,the carrier of M:];
      r ## p by A2,Th26;
      then r ## q by A1,Th21;
      hence thesis;
    end;
    thus x in q~ implies x in p~
    proof
      assume
A3:   x in q~;
      then reconsider
      r = x as Element of [:the carrier of M,the carrier of M:];
      r ## q by A3,Th26;
      then r ## p by A1,Th21;
      hence thesis;
    end;
  end;
  hence thesis by TARSKI:2;
end;
