
theorem
  for S being delta-concrete Categorial non empty Signature holds S is
  CatSignature of underlay S
proof
  let S be delta-concrete Categorial non empty Signature;
  set s = the SortSymbol of S;
  consider A being set such that
A1: CatSign A is Subsignature of S and
A2: the carrier of S = [:{0},2-tuples_on A:] by Def4;
  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
  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;
  consider i being (Element of NAT), p being FinSequence such that
A4: s = [i,p] and
A5: len p = f.i & [:{i}, (f.i)-tuples_on underlay S:] c= the carrier of
  S by A3;
  p in 2-tuples_on A by A2,A4,ZFMISC_1:105;
  then
A6: len p = 2 by FINSEQ_2:132;
A7: for x being object
    st x in proj2((the carrier of S) \/ (the carrier' of S))
     holds x is Function by Lm3;
A8: A c= underlay S
  proof
    let x be object;
    assume x in A;
    then <*x,x*> in 2-tuples_on A by FINSEQ_2:137;
    then [0,<*x,x*>] in the carrier of S by A2,ZFMISC_1:105;
    then
A9: [0,<*x,x*>] in (the carrier of S) \/ the carrier' of S by XBOOLE_0:def 3;
    rng <*x,x*> = {x,x} by FINSEQ_2:127;
    then x in rng <*x,x*> by TARSKI:def 2;
    hence thesis by A9,Def6,A7;
  end;
  i = 0 by A2,A4,ZFMISC_1:105;
  then
A10: 2-tuples_on underlay S c= 2-tuples_on A by A2,A5,A6,ZFMISC_1:94;
  underlay S c= A
  proof
    let x be object;
    assume x in underlay S;
    then <*x,x*> in 2-tuples_on underlay S by FINSEQ_2:137;
    hence thesis by A10,FINSEQ_2:138;
  end;
  then A = underlay S by A8;
  hence thesis by A1,A2,Def5;
end;
