
theorem Th53:
  for n being Nat for p being FinSeqLen of n for f being Function
  of n-tuples_on BOOLEAN, BOOLEAN holds 1GateCircStr(p,f) is gate`2isBoolean
proof
  set X = BOOLEAN;
  let n be Nat;
  let p be FinSeqLen of n;
  let f be Function of n-tuples_on X, X;
  let g be set;
A1: len p = n by CARD_1:def 7;
A2: (the Arity of 1GateCircStr(p,f)).[p,f] = p by Def6;
  assume g in the carrier' of 1GateCircStr(p,f);
  then
A3: g = [p,f] by Th41;
  let q be FinSequence;
  assume q = (the Arity of 1GateCircStr(p,f)).g;
  then reconsider f as Function of (len q)-tuples_on X, X by A3,A2,A1;
  take f;
  thus thesis by A3;
end;
