
theorem
  for S being one-gate ManySortedSign for A being one-gate Circuit of S
  for n being Element of NAT for X being finite non empty set for f being
  Function of n-tuples_on X, X, p being FinSeqLen of n st A = 1GateCircuit(p,f)
  holds S = 1GateCircStr(p,f)
proof
  let S be one-gate ManySortedSign;
  let A be one-gate Circuit of S;
  let n be Element of NAT;
  let X be finite non empty set;
  let f be Function of n-tuples_on X, X, p be FinSeqLen of n such that
A1: A = 1GateCircuit(p,f);
  consider X1 being non empty finite set, n1 being Element of NAT, p1 being
  FinSeqLen of n1, f1 being Function of n1-tuples_on X1,X1 such that
A2: S = 1GateCircStr(p1,f1) and
A3: A = 1GateCircuit(p1,f1) by Def7;
  {[p,f]} = the carrier' of 1GateCircStr(p,f) by CIRCCOMB:def 6
    .= dom the Charact of 1GateCircuit(p1,f1) by A1,A3,PARTFUN1:def 2
    .= the carrier' of 1GateCircStr(p1,f1) by PARTFUN1:def 2
    .= {[p1,f1]} by CIRCCOMB:def 6;
  then
A4: [p,f] = [p1,f1] by ZFMISC_1:3;
  then p = p1 by XTUPLE_0:1;
  hence thesis by A2,A4,XTUPLE_0:1;
end;
