
theorem Th35:
  for am,bm,cm,dm,cin being set holds [<*am,bm*>,xor2] in
  InnerVertices BitFTA3Str(am,bm,cm,dm,cin) & GFA3AdderOutput(am,bm,cm) in
InnerVertices BitFTA3Str(am,bm,cm,dm,cin) & [<*am,bm*>,nor2] in InnerVertices
BitFTA3Str(am,bm,cm,dm,cin) & [<*bm,cm*>,nor2] in InnerVertices BitFTA3Str(am,
bm,cm,dm,cin) & [<*cm,am*>,nor2] in InnerVertices BitFTA3Str(am,bm,cm,dm,cin)
& GFA3CarryOutput(am,bm,cm) in InnerVertices BitFTA3Str(am,bm,cm,dm,cin) & [<*
GFA3AdderOutput(am,bm,cm),cin*>,xor2] in InnerVertices BitFTA3Str(am,bm,cm,dm,
  cin) & GFA3AdderOutput(GFA3AdderOutput(am,bm,cm),cin,dm) in InnerVertices
  BitFTA3Str(am,bm,cm,dm,cin) & [<*GFA3AdderOutput(am,bm,cm),cin*>,nor2] in
InnerVertices BitFTA3Str(am,bm,cm,dm,cin) & [<*cin,dm*>,nor2] in InnerVertices
  BitFTA3Str(am,bm,cm,dm,cin) & [<*dm,GFA3AdderOutput(am,bm,cm)*>,nor2] in
InnerVertices BitFTA3Str(am,bm,cm,dm,cin) & GFA3CarryOutput(GFA3AdderOutput(am,
  bm,cm),cin,dm) in InnerVertices BitFTA3Str(am,bm,cm,dm,cin)
proof
  let am,bm,cm,dm,cin be set;
  set S = BitFTA3Str(am,bm,cm,dm,cin);
  set A1 = GFA3AdderOutput(am,bm,cm);
  set C1 = GFA3CarryOutput(am,bm,cm);
  set A2 = GFA3AdderOutput(A1,cin,dm);
  set C2 = GFA3CarryOutput(A1,cin,dm);
  set ambm0 = [<*am,bm*>, xor2 ];
  set ambm = [<*am,bm*>, nor2];
  set bmcm = [<*bm,cm*>, nor2];
  set cmam = [<*cm,am*>, nor2];
  set A1cin0 = [<*A1,cin*>,xor2 ];
  set A1cin = [<*A1,cin*>,nor2];
  set cindm = [<*cin,dm*>,nor2];
  set dmA1 = [<*dm,A1*>, nor2];
  set p1 = {ambm0,A1,ambm,bmcm,cmam,C1};
  set p2 = {A1cin0,A2,A1cin,cindm,dmA1,C2};
A1: ambm0 in p1 & A1 in p1 by ENUMSET1:def 4;
A2: ambm in p1 & bmcm in p1 by ENUMSET1:def 4;
A3: A1cin0 in p2 & A2 in p2 by ENUMSET1:def 4;
A4: cmam 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 = {ambm0,A1} \/ {ambm,bmcm,cmam,C1} \/ {A1cin0,A2} \/ {
  A1cin,cindm,dmA1,C2} by Th31
    .= 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;
