reserve MS for satisfying_equiv MusicStruct;
reserve a,b,c,d,e,f for Element of MS;

theorem Th28:
  for MS being satisfying_commutativity satisfying_equiv MusicStruct
  for a,b,c,d being Element of MS holds a,b equiv c,d iff b,a equiv d,c
  proof
    let MS be satisfying_commutativity satisfying_equiv MusicStruct;
    let a,b,c,d be Element of MS;
    hereby
      assume a,b equiv c,d;
      then (the Ratio of MS).(a,b) = (the Ratio of MS).(c,d) by Def08a;
      then (the Ratio of MS).(b,a) = (the Ratio of MS).(d,c) by Def11a;
      hence b,a equiv d,c by Def08a;
    end;
    assume b,a equiv d,c;
    then (the Ratio of MS).(b,a) = (the Ratio of MS).(d,c) by Def08a;
    then (the Ratio of MS).(a,b) = (the Ratio of MS).(c,d) by Def11a;
    hence a,b equiv c,d by Def08a;
  end;
