reserve x,y,z,c for object,
  f for Function of 2-tuples_on BOOLEAN, BOOLEAN;

theorem Th53:
  for X being finite non empty set, f being Function of 3
-tuples_on X, X for s being State of 1GateCircuit(<*x,y,z*>,f) holds Following
  s is stable
proof
  let X be finite non empty set, f be Function of 3-tuples_on X, X;
  set p = <*x,y,z*>;
  set S = 1GateCircStr(p,f);
  let s be State of 1GateCircuit(p,f);
  set s1 = Following s, s2 = Following s1;
A1: the carrier of S = rng p \/ {[p,f]} by CIRCCOMB:def 6
    .= {x,y,z} \/ {[p,f]} by FINSEQ_2:128;
A2: now
    let a be object;
A3: s1.z = s.z & s2.[p,f] = f.<*s1.x, s1.y, s1.z*> by Th52;
    assume a in the carrier of S;
    then a in {x,y,z} or a in {[p,f]} by A1,XBOOLE_0:def 3;
    then
A4: a = x or a = y or a = z or a = [p,f] by ENUMSET1:def 1,TARSKI:def 1;
    s1.x = s.x & s1.y = s.y by Th52;
    hence s2.a = s1.a by A4,A3,Th52;
  end;
  dom s1 = the carrier of S & dom s2 = the carrier of S by CIRCUIT1:3;
  hence Following s = Following Following s by A2,FUNCT_1:2;
end;
