
theorem
  for S being delta-concrete ManySortedSign, i being set, p1,p2 being
  FinSequence st len p2 = len p1 & rng p2 c= underlay S holds ([i,p1] in the
carrier of S implies [i,p2] in the carrier of S) & ([i,p1] in the carrier' of S
  implies [i,p2] in the carrier' of S)
proof
  let S be delta-concrete ManySortedSign, i be set, p1,p2 be FinSequence such
  that
A1: len p2 = len p1 & rng p2 c= underlay 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;
  hereby
    assume [i,p1] in the carrier of S;
    then consider j1 being (Element of NAT), q1 being FinSequence such that
A4: [i,p1] = [j1,q1] and
A5: len q1 = f.j1 and
A6: [:{j1}, (f.j1)-tuples_on underlay S:] c= the carrier of S by A2;
    p1 = q1 by A4,XTUPLE_0:1;
    then
A7: p2 in (f.j1)-tuples_on underlay S by A1,A5,FINSEQ_2:132;
    i = j1 by A4,XTUPLE_0:1;
    then [i,p2] in [:{j1}, (f.j1)-tuples_on underlay S:] by A7,ZFMISC_1:105;
    hence [i,p2] in the carrier of S by A6;
  end;
  assume [i,p1] in the carrier' of S;
  then consider j1 being (Element of NAT), q1 being FinSequence such that
A8: [i,p1] = [j1,q1] and
A9: len q1 = f.j1 and
A10: [:{j1}, (f.j1)-tuples_on underlay S:] c= the carrier' of S by A3;
  p1 = q1 by A8,XTUPLE_0:1;
  then
A11: p2 in (f.j1)-tuples_on underlay S by A1,A9,FINSEQ_2:132;
  i = j1 by A8,XTUPLE_0:1;
  then [i,p2] in [:{j1}, (f.j1)-tuples_on underlay S:] by A11,ZFMISC_1:105;
  hence thesis by A10;
end;
