theorem Th5:
  F is associative implies F/\/RD is associative
proof
  defpred P[Element of Class RD, Element of Class RD, Element of Class RD]
  means (F/\/RD).($1,(F/\/RD).($2,$3)) = (F/\/RD).((F/\/RD).($1,$2),$3);
  assume
A1: for d,a,b holds F.(d,F.(a,b)) = F.(F.(d,a),b);
A2: now
    let x1,x2,x3 be Element of D;
    (F/\/RD).(EqClass(RD,x1), (F/\/RD).(EqClass(RD,x2),EqClass(RD,x3))) =
    (F/\/RD).(Class(RD,x1),Class(RD, F.(x2,x3))) by Th3
      .= Class(RD, F.(x1,F.(x2,x3))) by Th3
      .= Class(RD, F.(F.(x1,x2),x3)) by A1
      .= (F/\/RD).(Class(RD,F.(x1,x2)),Class(RD, x3)) by Th3
      .= (F/\/RD).((F/\/RD).(EqClass(RD,x1),EqClass(RD,x2)), EqClass(RD,x3))
    by Th3;
    hence P[EqClass(RD,x1),EqClass(RD,x2),EqClass(RD,x3)];
  end;
  thus for c1,c2,c3 being Element of Class RD holds P[c1,c2,c3] from SchAux3(
  A2);
end;
