
theorem Th118:
  for x,y,z being set st x <> [<*y,z*>,nor2] & y <> [<*z,x*>,
  nor2] & z <> [<*x,y*>,nor2] for s being State of GFA3CarryCirc(x,y,z) holds
  Following(s,2) is stable
proof
  set f1 = nor2, f2 = nor2, f3 = nor2;
  let x,y,z be set such that
A1: x <> [<*y,z*>,f2] & y <> [<*z,x*>,f3] & z <> [<*x,y*>,f1];
  set S = GFA3CarryStr(x,y,z);
  reconsider xx = x, yy = y, zz = z as Vertex of S by Th111;
  let s be State of GFA3CarryCirc(x,y,z);
  set a1 = s.xx, a2 = s.yy, a3 = s.zz;
  set ffs = Following(s,2), fffs = Following ffs;
  set xy = [<*x,y*>,f1], yz = [<*y,z*>,f2], zx = [<*z,x*>,f3];
A2: ffs = Following Following s by FACIRC_1:15;
A3: z in InputVertices S by A1,Th113;
  then (Following s).z = a3 by CIRCUIT2:def 5;
  then
A4: ffs.z = a3 by A2,A3,CIRCUIT2:def 5;
A5: y in InputVertices S by A1,Th113;
  then (Following s).y = a2 by CIRCUIT2:def 5;
  then
A6: ffs.y = a2 by A2,A5,CIRCUIT2:def 5;
A7: x in InputVertices S by A1,Th113;
  then (Following s).x = a1 by CIRCUIT2:def 5;
  then
A8: ffs.x = a1 by A2,A7,CIRCUIT2:def 5;
  a3 = s.z;
  then
A9: ffs.xy = 'not' a1 '&' 'not' a2 by A1,Th117;
  a2 = s.y;
  then
A10: ffs.zx = 'not' a1 '&' 'not' a3 by A1,Th117;
  a1 = s.x;
  then
A11: ffs.yz = 'not' a2 '&' 'not' a3 by A1,Th117;
A12: ffs.GFA3CarryOutput(x,y,z) = 'not'(('not' a1 '&' 'not' a2) 'or' ('not'
  a2 '&' 'not' a3) 'or' ('not' a3 '&' 'not' a1)) by A1,Th117;
A13: now
    let a be object;
    assume
A14: a in the carrier of S;
    then reconsider v = a as Vertex of S;
A15: v in InputVertices S \/ InnerVertices S by A14,XBOOLE_1:45;
    thus ffs.a = (fffs).a
    proof
      per cases by A15,XBOOLE_0:def 3;
      suppose
        v in InputVertices S;
        hence thesis by CIRCUIT2:def 5;
      end;
      suppose
        v in InnerVertices S;
        then v in {xy, yz, zx} \/ {GFA3CarryOutput(x,y,z)} by Th106;
        then v in {xy, yz, zx} or v in {GFA3CarryOutput(x,y,z)} by
XBOOLE_0:def 3;
        then v = xy or v = yz or v = zx or v = GFA3CarryOutput(x,y,z) by
ENUMSET1:def 1,TARSKI:def 1;
        hence thesis by A12,A9,A11,A10,A8,A6,A4,Th115,Th116;
      end;
    end;
  end;
  dom Following Following(s,2) = the carrier of S & dom Following(s,2) =
  the carrier of S by CIRCUIT1:3;
  hence ffs = fffs by A13,FUNCT_1:2;
end;
