
theorem Th47:
  for x,b being non pair set for s being State of IncrementCirc(x,
b) holds (Following s).IncrementOutput(x,b) = and2a.<*s.x,s.b*> & (Following s)
  .x = s.x & (Following s).b = s.b
proof
  let x,b be non pair set;
  let s be State of IncrementCirc(x,b);
  set p = <*x,b*>;
  set S = IncrementStr(x,b);
A1: dom s = the carrier of S & x in the carrier of S by CIRCUIT1:3,FACIRC_1:43;
A2: b in the carrier of S by FACIRC_1:43;
  InnerVertices S = the carrier' of S by FACIRC_1:37;
  hence (Following s).IncrementOutput(x,b) = and2a.(s*p) by FACIRC_1:35
    .= and2a.<*s.x,s.b*> by A1,A2,FINSEQ_2:125;
  InputVertices S = {x,b} by FACIRC_1:40;
  then x in InputVertices S & b in InputVertices S by TARSKI:def 2;
  hence thesis by CIRCUIT2:def 5;
end;
