
theorem Th25:
  for am,bp,cm,dp,cin being set holds [<*am,bp*>,xor2c] in
  InnerVertices BitFTA2Str(am,bp,cm,dp,cin) & GFA2AdderOutput(am,bp,cm) in
InnerVertices BitFTA2Str(am,bp,cm,dp,cin) & [<*am,bp*>,and2a] in InnerVertices
BitFTA2Str(am,bp,cm,dp,cin) & [<*bp,cm*>,and2c] in InnerVertices BitFTA2Str(am,
bp,cm,dp,cin) & [<*cm,am*>,nor2] in InnerVertices BitFTA2Str(am,bp,cm,dp,cin)
& GFA2CarryOutput(am,bp,cm) in InnerVertices BitFTA2Str(am,bp,cm,dp,cin) & [<*
GFA2AdderOutput(am,bp,cm),cin*>,xor2c] in InnerVertices BitFTA2Str(am,bp,cm,dp,
  cin) & GFA1AdderOutput(GFA2AdderOutput(am,bp,cm),cin,dp) in InnerVertices
  BitFTA2Str(am,bp,cm,dp,cin) & [<*GFA2AdderOutput(am,bp,cm),cin*>,and2c] in
InnerVertices BitFTA2Str(am,bp,cm,dp,cin) & [<*cin,dp*>,and2a] in InnerVertices
  BitFTA2Str(am,bp,cm,dp,cin) & [<*dp,GFA2AdderOutput(am,bp,cm)*>,and2] in
InnerVertices BitFTA2Str(am,bp,cm,dp,cin) & GFA1CarryOutput(GFA2AdderOutput(am,
  bp,cm),cin,dp) in InnerVertices BitFTA2Str(am,bp,cm,dp,cin)
proof
  let am,bp,cm,dp,cin be set;
  set S = BitFTA2Str(am,bp,cm,dp,cin);
  set A1 = GFA2AdderOutput(am,bp,cm);
  set C1 = GFA2CarryOutput(am,bp,cm);
  set A2 = GFA1AdderOutput(A1,cin,dp);
  set C2 = GFA1CarryOutput(A1,cin,dp);
  set ambp0 = [<*am,bp*>, xor2c];
  set ambp = [<*am,bp*>, and2a];
  set bpcm = [<*bp,cm*>, and2c];
  set cmam = [<*cm,am*>, nor2];
  set A1cin0 = [<*A1,cin*>,xor2c];
  set A1cin = [<*A1,cin*>,and2c];
  set cindp = [<*cin,dp*>,and2a];
  set dpA1 = [<*dp,A1*>, and2 ];
  set p1 = {ambp0,A1,ambp,bpcm,cmam,C1};
  set p2 = {A1cin0,A2,A1cin,cindp,dpA1,C2};
A1: ambp0 in p1 & A1 in p1 by ENUMSET1:def 4;
A2: ambp in p1 & bpcm 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: dpA1 in p2 & C2 in p2 by ENUMSET1:def 4;
A6: A1cin in p2 & cindp in p2 by ENUMSET1:def 4;
  InnerVertices S = {ambp0,A1} \/ {ambp,bpcm,cmam,C1} \/ {A1cin0,A2} \/ {
  A1cin,cindp,dpA1,C2} by Th21
    .= p1 \/ {A1cin0,A2} \/ {A1cin,cindp,dpA1,C2} by ENUMSET1:12
    .= p1 \/ ({A1cin0,A2} \/ {A1cin,cindp,dpA1,C2}) by XBOOLE_1:4
    .= p1 \/ p2 by ENUMSET1:12;
  hence thesis by A1,A2,A4,A3,A6,A5,XBOOLE_0:def 3;
end;
