
theorem Th18:
  for A being non empty set, o being OperSymbol of CatSign A st o
  `1 = 2 or len o`2 = 3 ex a,b,c being Element of A st o = compsym(a,b,c)
proof
  let A be non empty set, o be OperSymbol of CatSign A such that
A1: o`1 = 2 or len o`2 = 3;
  the carrier' of CatSign A = [:{1},1-tuples_on A:] \/ [:{2},3-tuples_on A
  :] by Def3;
  then o in [:{1},1-tuples_on A:] or o in [:{2},3-tuples_on A:] by
XBOOLE_0:def 3;
  then
A2: o`1 in {1} & o`2 in 1-tuples_on A or o`1 in {2} & o`2 in 3-tuples_on A &
  o = [o`1,o`2] by MCART_1:10,21;
  then consider a,b,c being object such that
A3: a in A & b in A & c in A and
A4: o`2 = <*a,b,c*> by A1,CARD_1:def 7,FINSEQ_2:139,TARSKI:def 1;
  reconsider a,b,c as Element of A by A3;
  take a,b,c;
  thus thesis by A1,A2,A4,CARD_1:def 7,TARSKI:def 1;
end;
