
theorem
  for S being delta-concrete ManySortedSign, i being set, p1,p2 being
FinSequence st [i,p1] in the carrier of S & [i,p2] in the carrier of S or [i,p1
  ] in the carrier' of S & [i,p2] in the carrier' of S holds len p1 = len p2
proof
  let S be delta-concrete ManySortedSign, i be set, p1,p2 be FinSequence such
  that
A1: [i,p1] in the carrier of S & [i,p2] in the carrier of S or [i,p1] in
  the carrier' of S & [i,p2] in the carrier' of S;
  consider f being sequence of NAT such that
A2: for s being object st s in the carrier of S ex i being (Element of NAT)
  , p being FinSequence st s = [i,p] & len p = f.i & [:{i}, (f.i)-tuples_on
  underlay S:] c= the carrier of S and
A3: for o being object st o in the carrier' of S ex i being (Element of NAT
  ), p being FinSequence st o = [i,p] & len p = f.i & [:{i}, (f.i)-tuples_on
  underlay S:] c= the carrier' of S by Def7;
  per cases by A1;
  suppose
A4: [i,p1] in the carrier of S & [i,p2] in the carrier of S;
    then consider j1 being (Element of NAT), q1 being FinSequence such that
A5: [i,p1] = [j1,q1] and
A6: len q1 = f.j1 and
    [:{j1}, (f.j1)-tuples_on underlay S:] c= the carrier of S by A2;
A7: i = j1 & p1 = q1 by A5,XTUPLE_0:1;
    consider j2 being (Element of NAT), q2 being FinSequence such that
A8: [i,p2] = [j2,q2] and
A9: len q2 = f.j2 and
    [:{j2}, (f.j2)-tuples_on underlay S:] c= the carrier of S by A2,A4;
    i = j2 by A8,XTUPLE_0:1;
    hence thesis by A6,A8,A9,A7,XTUPLE_0:1;
  end;
  suppose
A10: [i,p1] in the carrier' of S & [i,p2] in the carrier' of S;
    then consider j1 being (Element of NAT), q1 being FinSequence such that
A11: [i,p1] = [j1,q1] and
A12: len q1 = f.j1 and
    [:{j1}, (f.j1)-tuples_on underlay S:] c= the carrier' of S by A3;
A13: i = j1 & p1 = q1 by A11,XTUPLE_0:1;
    consider j2 being (Element of NAT), q2 being FinSequence such that
A14: [i,p2] = [j2,q2] and
A15: len q2 = f.j2 and
    [:{j2}, (f.j2)-tuples_on underlay S:] c= the carrier' of S by A3,A10;
    i = j2 by A14,XTUPLE_0:1;
    hence thesis by A12,A14,A15,A13,XTUPLE_0:1;
  end;
end;
