
theorem
  for A1,A2 being set, S being CatSignature of A1 st S is CatSignature
  of A2 holds A1 = A2
proof
  let A1,A2 be set, S be CatSignature of A1;
  assume that
  CatSign A2 is Subsignature of S and
A1: the carrier of S = [:{0},2-tuples_on A2:];
A2: [:{0},2-tuples_on A1:] = [:{0},2-tuples_on A2:] by A1,Def5;
  then
A3: 2-tuples_on A1 c= 2-tuples_on A2 by ZFMISC_1:94;
  hereby
    let x be object;
    assume x in A1;
    then <*x,x*> in 2-tuples_on A1 by FINSEQ_2:137;
    then <*x,x*> in 2-tuples_on A2 by A3;
    hence x in A2 by FINSEQ_2:138;
  end;
  let x be object;
  assume x in A2;
  then
A4: <*x,x*> in 2-tuples_on A2 by FINSEQ_2:137;
  2-tuples_on A2 c= 2-tuples_on A1 by A2,ZFMISC_1:94;
  hence thesis by A4,FINSEQ_2:138;
end;
