
theorem Th2:
  for M being multMagma, R being Equivalence_Relation of M holds
  R is compatible iff for v,v9,w,w9 being Element of M
    st Class(R,v) = Class(R,v9) & Class(R,w) = Class(R,w9)
    holds Class(R,v*w) = Class(R,v9*w9)
proof
  let M be multMagma;
  let R be Equivalence_Relation of M;
  hereby
    assume A1: R is compatible;
    let v,v9,w,w9 be Element of M;
    assume A2: Class(R,v) = Class(R,v9) & Class(R,w) = Class(R,w9);
    per cases;
    suppose A3: M is empty;
      hence Class(R,v*w) = {} .= Class(R,v9*w9) by A3;
    end;
    suppose M is not empty; then
      v in Class(R,v9) & w in Class(R,w9) by A2,EQREL_1:23; then
      v*w in Class(R,v9*w9) by A1;
      hence Class(R,v*w) = Class(R,v9*w9) by EQREL_1:23;
    end;
  end;
  assume A4: for v,v9,w,w9 being Element of M
  st Class(R,v) = Class(R,v9) & Class(R,w) = Class(R,w9)
  holds Class(R,v*w) = Class(R,v9*w9);
  for v,v9,w,w9 being Element of M st v in Class(R,v9) & w in Class(R,w9)
  holds v*w in Class(R,v9*w9)
  proof
    let v,v9,w,w9 be Element of M;
    assume A5: v in Class(R,v9) & w in Class(R,w9);
    per cases;
    suppose M is empty; hence thesis by A5; end;
    suppose A6: M is not empty;
      Class(R,v9) = Class(R,v) &
      Class(R,w9) = Class(R,w) by A5,EQREL_1:23; then
      Class(R,v*w) = Class(R,v9*w9) by A4;
      hence v*w in Class(R,v9*w9) by A6,EQREL_1:23;
    end;
  end;
  hence R is compatible;
end;
