
theorem Th33:
  for am,bm,cm,dm being non pair set for cin being set st cin <> [
<*dm,GFA3AdderOutput(am,bm,cm)*>,nor2] & not cin in InnerVertices BitGFA3Str(
  am,bm,cm) holds InputVertices BitFTA3Str(am,bm,cm,dm,cin) = {am,bm,cm,dm,cin}
proof
  let am,bm,cm,dm be non pair set;
  let cin be set;
  set S = BitFTA3Str(am,bm,cm,dm,cin);
  set S1 = BitGFA3Str(am,bm,cm);
  set A1 = GFA3AdderOutput(am,bm,cm);
  set C1 = GFA3CarryOutput(am,bm,cm);
  set S2 = BitGFA3Str(A1,cin,dm);
  set ambm0 = [<*am,bm*>, xor2 ];
  set ambm = [<*am,bm*>, nor2];
  set bmcm = [<*bm,cm*>, nor2];
  set cmam = [<*cm,am*>, nor2];
  set cindm = [<*cin,dm*>,nor2];
  set dmA1 = [<*dm,A1*>, nor2];
  assume that
A1: cin <> dmA1 and
A2: not cin in InnerVertices S1;
A3: not dm in {A1,ambm0,ambm,bmcm,cmam,C1} by ENUMSET1:def 4;
  A1 in {A1,ambm0,ambm,bmcm,cmam,C1} by ENUMSET1:def 4;
  then
A4: {A1} \ {A1,ambm0,ambm,bmcm,cmam,C1} = {} by ZFMISC_1:60;
A5: InnerVertices S1 = {ambm0} \/ {A1} \/ {ambm,bmcm,cmam} \/ {C1} by
GFACIRC1:127
    .= {ambm0,A1} \/ {ambm,bmcm,cmam} \/ {C1} by ENUMSET1:1
    .= {ambm0,A1} \/ ({ambm,bmcm,cmam} \/ {C1}) by XBOOLE_1:4
    .= {A1,ambm0} \/ {ambm,bmcm,cmam,C1} by ENUMSET1:6
    .= {A1,ambm0,ambm,bmcm,cmam,C1} by ENUMSET1:12;
  then
A6: {A1,cin,dm} \ InnerVertices S1 = ({A1} \/ {cin,dm}) \ {A1,ambm0,ambm,
  bmcm,cmam,C1} by ENUMSET1:2
    .= ({A1} \ {A1,ambm0,ambm,bmcm,cmam,C1}) \/ ({cin,dm} \ {A1,ambm0,ambm,
  bmcm,cmam,C1}) by XBOOLE_1:42
    .= ({cin} \/ {dm}) \ {A1,ambm0,ambm,bmcm,cmam,C1} by A4,ENUMSET1:1
    .= ({cin} \ {A1,ambm0,ambm,bmcm,cmam,C1}) \/ ({dm} \ {A1,ambm0,ambm,bmcm
  ,cmam,C1}) by XBOOLE_1:42
    .= ({cin}) \/ ({dm} \ {A1,ambm0,ambm,bmcm,cmam,C1}) by A2,A5,ZFMISC_1:59
    .= ({cin}) \/ ({dm}) by A3,ZFMISC_1:59
    .= {cin,dm} by ENUMSET1:1;
A7: A1 <> cindm by Lm31;
  InnerVertices S2 misses InputVertices S1 & S1 tolerates S2 by Lm32,
CIRCCOMB:47;
  hence InputVertices (S) = (InputVertices S1) \/ (InputVertices S2 \
  InnerVertices S1) by FACIRC_1:4
    .= ({am,bm,cm}) \/ (InputVertices S2 \ InnerVertices S1) by GFACIRC1:130
    .= ({am,bm,cm}) \/ ({A1,cin,dm} \ InnerVertices S1) by A1,A7,GFACIRC1:129
    .= {am,bm,cm,dm,cin} by A6,ENUMSET1:9;
end;
