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

theorem
  the Equidistance of MS is Equivalence_Relation of
  [:the carrier of MS,the carrier of MS:]
  proof
    set R = the Equidistance of MS,
    C = [:the carrier of MS,the carrier of MS:];
    now
      dom R = C by Th25,TAXONOM1:3;
      hence R is total by PARTFUN1:def 2;
      field R = C & R is_symmetric_in C by Th25,Th26,PARTIT_2:9;
      hence R is symmetric by RELAT_2:def 11;
      field R = C & R is_transitive_in C by Th25,Th27,PARTIT_2:9;
      hence R is transitive by RELAT_2:def 16;
    end;
    hence thesis;
  end;
