
theorem Th15:
  for A being non empty set, S being CatSignature of A for s being
  SortSymbol of S ex a,b being Element of A st s = homsym(a,b)
proof
  let A be non empty set, S be CatSignature of A;
  let s be SortSymbol of S;
A1: the carrier of S = [:{0},2-tuples_on A:] by Def5;
  then s`2 in 2-tuples_on A by MCART_1:10;
  then s`2 in {z where z is Element of A*: len z = 2} by FINSEQ_2:def 4;
  then consider z being Element of A* such that
A2: s`2 = z and
A3: len z = 2;
A4: z.1 in {z.1,z.2} & z.2 in {z.1,z.2} by TARSKI:def 2;
A5: z = <*z.1,z.2*> by A3,FINSEQ_1:44;
  then rng z = {z.1,z.2} by FINSEQ_2:127;
  then reconsider a = z.1, b = z.2 as Element of A by A4;
  take a,b;
  s = [s`1,s`2] & s`1 in {0} by A1,MCART_1:10,21;
  hence thesis by A2,A5,TARSKI:def 1;
end;
