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

theorem Th48:
  for X being finite non empty set, f being Function of 2
-tuples_on X, X for s being State of 1GateCircuit(<*x,y*>,f) holds (Following s
).[<*x,y*>, f] = f.<*s.x, s.y*> & (Following s).x = s.x & (Following s).y = s.y
proof
  let X be finite non empty set, f be Function of 2-tuples_on X, X;
  let s be State of 1GateCircuit(<*x,y*>,f);
  set p = <*x,y*>;
  dom s = the carrier of 1GateCircStr(p, f) by CIRCUIT1:3;
  then
A1: dom s = rng p \/ {[p,f]} by CIRCCOMB:def 6;
  y in {x,y} by TARSKI:def 2;
  then y in rng p by FINSEQ_2:127;
  then
A2: y in dom s by A1,XBOOLE_0:def 3;
  x in {x,y} by TARSKI:def 2;
  then x in rng p by FINSEQ_2:127;
  then
A3: x in dom s by A1,XBOOLE_0:def 3;
  thus (Following s).[<*x,y*>, f] = f.(s*<*x,y*>) by CIRCCOMB:56
    .= f.<*s.x, s.y*> by A3,A2,FINSEQ_2:125;
  reconsider x, y as Vertex of 1GateCircStr(p,f) by Th43;
  InputVertices 1GateCircStr(p,f) = rng p by CIRCCOMB:42
    .= {x,y} by FINSEQ_2:127;
  then
  x in InputVertices 1GateCircStr(p,f) & y in InputVertices 1GateCircStr(
  p,f) by TARSKI:def 2;
  hence thesis by CIRCUIT2:def 5;
end;
