
theorem
  for x,b being non pair set for s being State of BitCompCirc(x,b) holds
  (Following s) is stable
proof
  let x,b be non pair set;
  set p = <*x,b*>;
  set S = BitCompStr(x,b);
  let s be State of BitCompCirc(x,b);
  set s1 = Following s, s2 = Following s1;
A1: the carrier of S = {x,b} \/ {[p,xor2a],[p,and2a]} by Th37;
A2: now
    let a be object;
A3: s1.[p,xor2a] = s1.CompOutput(x,b) .= xor2a.<*s.x, s.b*> by Th45;
    assume a in the carrier of S;
    then a in {x,b} or a in {[p,xor2a],[p,and2a]} by A1,XBOOLE_0:def 3;
    then
A4: a = x or a = b or a = [p,xor2a] or a = [p,and2a] by TARSKI:def 2;
A5: s2.[p,and2a] = s2.IncrementOutput(x,b) .= and2a.<*s1.x, s1.b*> by Th49;
A6: s2.[p,xor2a] = s2.CompOutput(x,b) .= xor2a.<*s1.x, s1.b*> by Th45;
A7: s1.[p,and2a] = s1.IncrementOutput(x,b) .= and2a.<*s.x, s.b*> by Th49;
    s1.x = s.x by Th45;
    hence s2.a = s1.a by A4,A3,A7,A6,A5,Th45;
  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;
