
theorem Th40:
  for f being object, p being FinSequence holds the Arity of
1GateCircStr(p,f) = (p,f) .--> p & the ResultSort of 1GateCircStr(p,f) = (p,f)
  .--> [p,f]
proof
  let f be object, p be FinSequence;
  set S = 1GateCircStr(p,f);
  (the Arity of S).[p,f] = p by Def6;
  then
A1: for x being object st x in {[p,f]} holds (the Arity of S).x = p by
TARSKI:def 1;
A2: the carrier' of S = {[p,f]} by Def6;
  then dom the Arity of S = {[p,f]} by FUNCT_2:def 1;
  hence the Arity of 1GateCircStr(p,f) = (p,f) .--> p by A1,FUNCOP_1:11;
  (the ResultSort of S).[p,f] = [p,f] by Def6;
  then
A3: for x being object st x in {[p,f]} holds (the ResultSort of S).x = [p,f]
    by TARSKI:def 1;
  dom the ResultSort of S = {[p,f]} by A2,FUNCT_2:def 1;
  hence thesis by A3,FUNCOP_1:11;
end;
