
theorem Th13:
  for ap,bm,cp,dm being non pair set for cin being set st cin <> [
<*dm,GFA1AdderOutput(ap,bm,cp)*>,nor2] & not cin in InnerVertices BitGFA1Str(
  ap,bm,cp) holds InputVertices BitFTA1Str(ap,bm,cp,dm,cin) = {ap,bm,cp,dm,cin}
proof
  let ap,bm,cp,dm be non pair set;
  let cin be set;
  set S = BitFTA1Str(ap,bm,cp,dm,cin);
  set S1 = BitGFA1Str(ap,bm,cp);
  set A1 = GFA1AdderOutput(ap,bm,cp);
  set C1 = GFA1CarryOutput(ap,bm,cp);
  set S2 = BitGFA2Str(A1,cin,dm);
  set apbm0 = [<*ap,bm*>, xor2c];
  set apbm = [<*ap,bm*>, and2c];
  set bmcp = [<*bm,cp*>, and2a];
  set cpap = [<*cp,ap*>, and2 ];
  set dmA1 = [<*dm,A1*>, nor2];
  assume that
A1: cin <> dmA1 and
A2: not cin in InnerVertices S1;
A3: not dm in {A1,apbm0,apbm,bmcp,cpap,C1} by ENUMSET1:def 4;
  A1 in {A1,apbm0,apbm,bmcp,cpap,C1} by ENUMSET1:def 4;
  then
A4: {A1} \ {A1,apbm0,apbm,bmcp,cpap,C1} = {} by ZFMISC_1:60;
A5: InnerVertices S1 = {apbm0} \/ {A1} \/ {apbm,bmcp,cpap} \/ {C1} by
GFACIRC1:63
    .= {apbm0,A1} \/ {apbm,bmcp,cpap} \/ {C1} by ENUMSET1:1
    .= {apbm0,A1} \/ ({apbm,bmcp,cpap} \/ {C1}) by XBOOLE_1:4
    .= {A1,apbm0} \/ {apbm,bmcp,cpap,C1} by ENUMSET1:6
    .= {A1,apbm0,apbm,bmcp,cpap,C1} by ENUMSET1:12;
  then
A6: {A1,cin,dm} \ InnerVertices S1 = ({A1} \/ {cin,dm}) \ {A1,apbm0,apbm,
  bmcp,cpap,C1} by ENUMSET1:2
    .= ({A1} \ {A1,apbm0,apbm,bmcp,cpap,C1}) \/ ({cin,dm} \ {A1,apbm0,apbm,
  bmcp,cpap,C1}) by XBOOLE_1:42
    .= ({cin} \/ {dm}) \ {A1,apbm0,apbm,bmcp,cpap,C1} by A4,ENUMSET1:1
    .= ({cin} \ {A1,apbm0,apbm,bmcp,cpap,C1}) \/ ({dm} \ {A1,apbm0,apbm,bmcp
  ,cpap,C1}) by XBOOLE_1:42
    .= ({cin}) \/ ({dm} \ {A1,apbm0,apbm,bmcp,cpap,C1}) by A2,A5,ZFMISC_1:59
    .= ({cin}) \/ ({dm}) by A3,ZFMISC_1:59
    .= {cin,dm} by ENUMSET1:1;
  InnerVertices S2 misses InputVertices S1 & S1 tolerates S2 by Lm12,
CIRCCOMB:47;
  hence InputVertices (S) = (InputVertices S1) \/ (InputVertices S2 \
  InnerVertices S1) by FACIRC_1:4
    .= ({ap,bm,cp}) \/ (InputVertices S2 \ InnerVertices S1) by GFACIRC1:66
    .= ({ap,bm,cp}) \/ ({A1,cin,dm} \ InnerVertices S1) by A1,Lm11,GFACIRC1:97
    .= {ap,bm,cp,dm,cin} by A6,ENUMSET1:9;
end;
