
theorem Th11:
  for A being non empty set, S being CatSignature of A for a being
  Element of A holds idsym a in the carrier' of S & for b being Element of A
holds homsym(a,b) in the carrier of S & for c being Element of A holds compsym(
  a,b,c) in the carrier' of S
proof
  let A be non empty set, S be CatSignature of A;
  let a be Element of A;
A1: the carrier' of CatSign A = [:{1},1-tuples_on A:] \/ [:{2},3-tuples_on A
  :] by Def3;
A2: CatSign A is Subsignature of S by Def5;
  then
A3: the carrier of CatSign A c= the carrier of S by INSTALG1:10;
A4: the carrier' of CatSign A c= the carrier' of S by A2,INSTALG1:10;
  <*a*> in 1-tuples_on A by FINSEQ_2:135;
  then [1,<*a*>] in [:{1},1-tuples_on A:] by ZFMISC_1:105;
  then [1,<*a*>] in the carrier' of CatSign A by A1,XBOOLE_0:def 3;
  hence idsym a in the carrier' of S by A4;
  let b be Element of A;
A5: the carrier of CatSign A = [:{0},2-tuples_on A:] by Def3;
  <*a,b*> in 2-tuples_on A by FINSEQ_2:137;
  then [0,<*a,b*>] in [:{0},2-tuples_on A:] by ZFMISC_1:105;
  hence homsym(a,b) in the carrier of S by A3,A5;
  let c be Element of A;
  <*a,b,c*> in 3-tuples_on A by FINSEQ_2:139;
  then [2,<*a,b,c*>] in [:{2},3-tuples_on A:] by ZFMISC_1:105;
  then [2,<*a,b,c*>] in the carrier' of CatSign A by A1,XBOOLE_0:def 3;
  hence thesis by A4;
end;
