
theorem Th52:
  for n being Nat, X being non empty set for f being Function of n
  -tuples_on X, X for p being FinSeqLen of n holds 1GateCircuit(p,f) is
  gate`2=den & 1GateCircStr(p,f) is gate`2=den
proof
  let n be Nat;
  let X be non empty set;
  let f be Function of n-tuples_on X, X;
  let p be FinSeqLen of n;
  thus
A1: 1GateCircuit(p,f) is gate`2=den
  proof
    let g be set;
    assume g in the carrier' of 1GateCircStr(p,f);
    then
A2: g = [p,f] by Th41;
    hence g = [g`1, f]
      .= [g`1, (the Charact of 1GateCircuit(p,f)).g] by A2,Def13;
  end;
  take 1GateCircuit(p,f);
  thus thesis by A1;
end;
