
theorem Th11:
  for x,y,z being set holds InnerVertices GFA0CarryStr(x,y,z) = {[
  <*x,y*>,and2], [<*y,z*>,and2], [<*z,x*>,and2]} \/ {GFA0CarryOutput(x,y,z)}
proof
  let x,y,z be set;
  set f1 = and2, f2 = and2, f3 = and2, f4 = or3;
  set xy = [<*x,y*>,f1], yz = [<*y,z*>,f2], zx = [<*z,x*>,f3];
  set Cxy = 1GateCircStr(<*x,y*>,f1);
  set Cyz = 1GateCircStr(<*y,z*>,f2);
  set Czx = 1GateCircStr(<*z,x*>,f3);
  set Cxyz = 1GateCircStr(<*xy, yz, zx*>,f4);
A1: Cxy tolerates (Cyz +* Czx +* Cxyz) by CIRCCOMB:47;
  Cyz tolerates (Czx +* Cxyz) by CIRCCOMB:47;
  then
A2: InnerVertices (Cyz +* (Czx +* Cxyz)) = InnerVertices Cyz \/
  InnerVertices (Czx +* Cxyz) by CIRCCOMB:11;
  Czx tolerates Cxyz by CIRCCOMB:47;
  then
A3: InnerVertices (Czx +* Cxyz) = InnerVertices Czx \/ InnerVertices Cxyz by
CIRCCOMB:11;
  thus InnerVertices GFA0CarryStr(x,y,z) = InnerVertices (Cxy +* (Cyz +* Czx)
  +* Cxyz) by CIRCCOMB:6
    .= InnerVertices (Cxy +* (Cyz +* Czx +* Cxyz)) by CIRCCOMB:6
    .= InnerVertices Cxy \/ InnerVertices (Cyz +* Czx +* Cxyz) by A1,
CIRCCOMB:11
    .= InnerVertices Cxy \/ InnerVertices (Cyz +* (Czx +* Cxyz)) by CIRCCOMB:6
    .= InnerVertices Cxy \/ InnerVertices Cyz \/ (InnerVertices Czx \/
  InnerVertices Cxyz) by A2,A3,XBOOLE_1:4
    .= InnerVertices Cxy \/ InnerVertices Cyz \/ InnerVertices Czx \/
  InnerVertices Cxyz by XBOOLE_1:4
    .= {xy} \/ InnerVertices Cyz \/ InnerVertices Czx \/ InnerVertices Cxyz
  by CIRCCOMB:42
    .= {xy} \/ {yz} \/ InnerVertices Czx \/ InnerVertices Cxyz by CIRCCOMB:42
    .= {xy} \/ {yz} \/ {zx} \/ InnerVertices Cxyz by CIRCCOMB:42
    .= {xy, yz} \/ {zx} \/ InnerVertices Cxyz by ENUMSET1:1
    .= {xy, yz, zx} \/ InnerVertices Cxyz by ENUMSET1:3
    .= {xy, yz, zx} \/ {GFA0CarryOutput(x,y,z)} by CIRCCOMB:42;
end;
