
theorem Th15:
  for ap,bm,cp,dm,cin being set holds [<*ap,bm*>,xor2c] in
  InnerVertices BitFTA1Str(ap,bm,cp,dm,cin) & GFA1AdderOutput(ap,bm,cp) in
InnerVertices BitFTA1Str(ap,bm,cp,dm,cin) & [<*ap,bm*>,and2c] in InnerVertices
BitFTA1Str(ap,bm,cp,dm,cin) & [<*bm,cp*>,and2a] in InnerVertices BitFTA1Str(ap,
bm,cp,dm,cin) & [<*cp,ap*>,and2 ] in InnerVertices BitFTA1Str(ap,bm,cp,dm,cin)
& GFA1CarryOutput(ap,bm,cp) in InnerVertices BitFTA1Str(ap,bm,cp,dm,cin) & [<*
GFA1AdderOutput(ap,bm,cp),cin*>,xor2c] in InnerVertices BitFTA1Str(ap,bm,cp,dm,
  cin) & GFA2AdderOutput(GFA1AdderOutput(ap,bm,cp),cin,dm) in InnerVertices
  BitFTA1Str(ap,bm,cp,dm,cin) & [<*GFA1AdderOutput(ap,bm,cp),cin*>,and2a] in
InnerVertices BitFTA1Str(ap,bm,cp,dm,cin) & [<*cin,dm*>,and2c] in InnerVertices
  BitFTA1Str(ap,bm,cp,dm,cin) & [<*dm,GFA1AdderOutput(ap,bm,cp)*>,nor2] in
InnerVertices BitFTA1Str(ap,bm,cp,dm,cin) & GFA2CarryOutput(GFA1AdderOutput(ap,
  bm,cp),cin,dm) in InnerVertices BitFTA1Str(ap,bm,cp,dm,cin)
proof
  let ap,bm,cp,dm,cin be set;
  set S = BitFTA1Str(ap,bm,cp,dm,cin);
  set A1 = GFA1AdderOutput(ap,bm,cp);
  set C1 = GFA1CarryOutput(ap,bm,cp);
  set A2 = GFA2AdderOutput(A1,cin,dm);
  set C2 = GFA2CarryOutput(A1,cin,dm);
  set apbm0 = [<*ap,bm*>, xor2c];
  set apbm = [<*ap,bm*>, and2c];
  set bmcp = [<*bm,cp*>, and2a];
  set cpap = [<*cp,ap*>, and2 ];
  set A1cin0 = [<*A1,cin*>,xor2c];
  set A1cin = [<*A1,cin*>,and2a];
  set cindm = [<*cin,dm*>,and2c];
  set dmA1 = [<*dm,A1*>, nor2];
  set p1 = {apbm0,A1,apbm,bmcp,cpap,C1};
  set p2 = {A1cin0,A2,A1cin,cindm,dmA1,C2};
A1: apbm0 in p1 & A1 in p1 by ENUMSET1:def 4;
A2: apbm in p1 & bmcp in p1 by ENUMSET1:def 4;
A3: A1cin0 in p2 & A2 in p2 by ENUMSET1:def 4;
A4: cpap in p1 & C1 in p1 by ENUMSET1:def 4;
A5: dmA1 in p2 & C2 in p2 by ENUMSET1:def 4;
A6: A1cin in p2 & cindm in p2 by ENUMSET1:def 4;
  InnerVertices S = {apbm0,A1} \/ {apbm,bmcp,cpap,C1} \/ {A1cin0,A2} \/ {
  A1cin,cindm,dmA1,C2} by Th11
    .= p1 \/ {A1cin0,A2} \/ {A1cin,cindm,dmA1,C2} by ENUMSET1:12
    .= p1 \/ ({A1cin0,A2} \/ {A1cin,cindm,dmA1,C2}) by XBOOLE_1:4
    .= p1 \/ p2 by ENUMSET1:12;
  hence thesis by A1,A2,A4,A3,A6,A5,XBOOLE_0:def 3;
end;
