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

theorem Th27:
  the Equidistance of MS is_transitive_in
    [: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
      let x,y,z be object;
      assume that
A1:   x in C and
A2:   y in C and
A3:   z in C and
A4:   [x,y] in R and
A5:   [y,z] in R;
      consider x1,x2 be object such that
A6:   x1 in the carrier of MS and
A7:   x2 in the carrier of MS and
A8:   x = [x1,x2] by A1,ZFMISC_1:def 2;
      consider y1,y2 be object such that
A9:   y1 in the carrier of MS and
A10:  y2 in the carrier of MS and
A11:  y = [y1,y2] by A2,ZFMISC_1:def 2;
      consider z1,z2 be object such that
A12:  z1 in the carrier of MS and
A13:  z2 in the carrier of MS and
A14:  z = [z1,z2] by A3,ZFMISC_1:def 2;
      reconsider x1,x2,y1,y2,z1,z2 as Element of MS
        by A6,A7,A9,A10,A12,A13;
      x1,x2 equiv y1,y2 & y1,y2 equiv z1,z2 by A4,A5,A8,A11,A14;
      then x1,x2 equiv z1,z2 by Th23;
      hence [x,z] in R by A8,A14;
    end;
    hence thesis by RELAT_2:def 8;
  end;
