
theorem Th45:
  for x,b being non pair set for s being State of BitCompCirc(x,b)
holds (Following s).CompOutput(x,b) = xor2a.<*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 BitCompCirc(x,b);
  set p = <*x,b*>;
  set S = BitCompStr(x,b);
A1: dom s = the carrier of S & x in the carrier of S by Th36,CIRCUIT1:3;
A2: b in the carrier of S by Th36;
  InnerVertices S = the carrier' of S by FACIRC_1:37;
  hence (Following s).CompOutput(x,b) = xor2a.(s*p) by Th39,FACIRC_1:35
    .= xor2a.<*s.x,s.b*> by A1,A2,FINSEQ_2:125;
  x in InputVertices S & b in InputVertices S by Th41;
  hence thesis by CIRCUIT2:def 5;
end;
