
theorem Th23:
  for am,bp,cm,dp being non pair set for cin being set st cin <> [
<*dp,GFA2AdderOutput(am,bp,cm)*>,and2] & not cin in InnerVertices BitGFA2Str(am
  ,bp,cm) holds InputVertices BitFTA2Str(am,bp,cm,dp,cin) = {am,bp,cm,dp,cin}
proof
  let am,bp,cm,dp be non pair set;
  let cin be set;
  set S = BitFTA2Str(am,bp,cm,dp,cin);
  set S1 = BitGFA2Str(am,bp,cm);
  set A1 = GFA2AdderOutput(am,bp,cm);
  set C1 = GFA2CarryOutput(am,bp,cm);
  set S2 = BitGFA1Str(A1,cin,dp);
  set ambp0 = [<*am,bp*>, xor2c];
  set ambp = [<*am,bp*>, and2a];
  set bpcm = [<*bp,cm*>, and2c];
  set cmam = [<*cm,am*>, nor2];
  set dpA1 = [<*dp,A1*>, and2 ];
  assume that
A1: cin <> dpA1 and
A2: not cin in InnerVertices S1;
A3: not dp in {A1,ambp0,ambp,bpcm,cmam,C1} by ENUMSET1:def 4;
  A1 in {A1,ambp0,ambp,bpcm,cmam,C1} by ENUMSET1:def 4;
  then
A4: {A1} \ {A1,ambp0,ambp,bpcm,cmam,C1} = {} by ZFMISC_1:60;
A5: InnerVertices S1 = {ambp0} \/ {A1} \/ {ambp,bpcm,cmam} \/ {C1} by
GFACIRC1:95
    .= {ambp0,A1} \/ {ambp,bpcm,cmam} \/ {C1} by ENUMSET1:1
    .= {ambp0,A1} \/ ({ambp,bpcm,cmam} \/ {C1}) by XBOOLE_1:4
    .= {A1,ambp0} \/ {ambp,bpcm,cmam,C1} by ENUMSET1:6
    .= {A1,ambp0,ambp,bpcm,cmam,C1} by ENUMSET1:12;
  then
A6: {A1,cin,dp} \ InnerVertices S1 = ({A1} \/ {cin,dp}) \ {A1,ambp0,ambp,
  bpcm,cmam,C1} by ENUMSET1:2
    .= ({A1} \ {A1,ambp0,ambp,bpcm,cmam,C1}) \/ ({cin,dp} \ {A1,ambp0,ambp,
  bpcm,cmam,C1}) by XBOOLE_1:42
    .= ({cin} \/ {dp}) \ {A1,ambp0,ambp,bpcm,cmam,C1} by A4,ENUMSET1:1
    .= ({cin} \ {A1,ambp0,ambp,bpcm,cmam,C1}) \/ ({dp} \ {A1,ambp0,ambp,bpcm
  ,cmam,C1}) by XBOOLE_1:42
    .= ({cin}) \/ ({dp} \ {A1,ambp0,ambp,bpcm,cmam,C1}) by A2,A5,ZFMISC_1:59
    .= ({cin}) \/ ({dp}) by A3,ZFMISC_1:59
    .= {cin,dp} by ENUMSET1:1;
  InnerVertices S2 misses InputVertices S1 & S1 tolerates S2 by Lm22,
CIRCCOMB:47;
  hence InputVertices (S) = (InputVertices S1) \/ (InputVertices S2 \
  InnerVertices S1) by FACIRC_1:4
    .= ({am,bp,cm}) \/ (InputVertices S2 \ InnerVertices S1) by GFACIRC1:98
    .= ({am,bp,cm}) \/ ({A1,cin,dp} \ InnerVertices S1) by A1,Lm21,GFACIRC1:65
    .= {am,bp,cm,dp,cin} by A6,ENUMSET1:9;
end;
