
theorem Th7:
  for S being delta-concrete ManySortedSign, x being set st x in
the carrier of S or x in the carrier' of S ex i being (Element of NAT), p being
  FinSequence st x = [i,p] & rng p c= underlay S
proof
  let S be delta-concrete ManySortedSign, x be set such that
A1: x in the carrier of S or x in the carrier' of S;
A2: x in (the carrier of S) \/ the carrier' of S by A1,XBOOLE_0:def 3;
  consider f being sequence of NAT such that
A3: 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
A4: 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;
A5: for x being object
    st x in proj2((the carrier of S) \/ (the carrier' of S))
     holds x is Function by Lm3;
  per cases by A1;
  suppose
    x in the carrier of S;
    then consider i being (Element of NAT), p being FinSequence such that
A6: x = [i,p] and
    len p = f.i and
    [:{i}, (f.i)-tuples_on underlay S:] c= the carrier of S by A3;
    take i,p;
    thus x = [i,p] by A6;
    let a be object;
    thus thesis by A2,A6,Def6,A5;
  end;
  suppose
    x in the carrier' of S;
    then consider i being (Element of NAT), p being FinSequence such that
A7: x = [i,p] and
    len p = f.i and
    [:{i}, (f.i)-tuples_on underlay S:] c= the carrier' of S by A4;
    take i,p;
    thus x = [i,p] by A7;
    let a be object;
    thus thesis by A2,A7,Def6,A5;
  end;
end;
