
theorem
  for A being set holds underlay CatSign A = A
proof
  let A be set;
  set S = CatSign A;
A1: the carrier' of S = [:{1},1-tuples_on A:] \/ [:{2}, 3-tuples_on A:]
     by Def3;
A2: for x being object
    st x in proj2((the carrier of CatSign A) \/ (the carrier' of CatSign A))
     holds x is Function by Lm2;
  hereby
    let x be object;
    assume x in underlay CatSign A;
    then consider a being set, f being Function such that
A3: [a,f] in (the carrier of S) \/ (the carrier' of S) and
A4: x in rng f by Def6,A2;
    [a,f] in the carrier of S or [a,f] in the carrier' of S by A3,
XBOOLE_0:def 3;
    then
    [a,f] in [:{0},2-tuples_on A:] or [a,f] in [:{1},1-tuples_on A:] or [a
    ,f] in [:{2},3-tuples_on A:] by A1,Def3,XBOOLE_0:def 3;
    then f in 2-tuples_on A or f in 1-tuples_on A or f in 3-tuples_on A by
ZFMISC_1:87;
    then f is FinSequence of A by FINSEQ_2:def 3;
    then rng f c= A by FINSEQ_1:def 4;
    hence x in A by A4;
  end;
  let x be object;
  assume x in A;
  then
A5: <*x,x*> in 2-tuples_on A by FINSEQ_2:137;
  the carrier of S = [:{0},2-tuples_on A:] by Def3;
  then [0,<*x,x*>] in the carrier of S by A5,ZFMISC_1:105;
  then
A6: [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 A6,Def6,A2;
end;
