
theorem Th3:
  for ap,bp,cp,dp being non pair set for cin being set st cin <> [
<*dp,GFA0AdderOutput(ap,bp,cp)*>,and2] & not cin in InnerVertices BitGFA0Str(ap
  ,bp,cp) holds InputVertices BitFTA0Str(ap,bp,cp,dp,cin) = {ap,bp,cp,dp,cin}
proof
  let ap,bp,cp,dp be non pair set;
  let cin be set;
  set S = BitFTA0Str(ap,bp,cp,dp,cin);
  set S1 = BitGFA0Str(ap,bp,cp);
  set A1 = GFA0AdderOutput(ap,bp,cp);
  set C1 = GFA0CarryOutput(ap,bp,cp);
  set S2 = BitGFA0Str(A1,cin,dp);
  set apbp0 = [<*ap,bp*>, xor2];
  set apbp = [<*ap,bp*>, and2];
  set bpcp = [<*bp,cp*>, and2];
  set cpap = [<*cp,ap*>, and2];
  set cindp = [<*cin,dp*>,and2];
  set dpA1 = [<*dp,A1*>, and2];
  assume that
A1: cin <> dpA1 and
A2: not cin in InnerVertices S1;
A3: not dp in {A1,apbp0,apbp,bpcp,cpap,C1} by ENUMSET1:def 4;
  A1 in {A1,apbp0,apbp,bpcp,cpap,C1} by ENUMSET1:def 4;
  then
A4: {A1} \ {A1,apbp0,apbp,bpcp,cpap,C1} = {} by ZFMISC_1:60;
A5: InnerVertices S1 = {apbp0} \/ {A1} \/ {apbp,bpcp,cpap} \/ {C1} by
GFACIRC1:31
    .= {apbp0,A1} \/ {apbp,bpcp,cpap} \/ {C1} by ENUMSET1:1
    .= {apbp0,A1} \/ ({apbp,bpcp,cpap} \/ {C1}) by XBOOLE_1:4
    .= {A1,apbp0} \/ {apbp,bpcp,cpap,C1} by ENUMSET1:6
    .= {A1,apbp0,apbp,bpcp,cpap,C1} by ENUMSET1:12;
  then
A6: {A1,cin,dp} \ InnerVertices S1 = ({A1} \/ {cin,dp}) \ {A1,apbp0,apbp,
  bpcp,cpap,C1} by ENUMSET1:2
    .= ({A1} \ {A1,apbp0,apbp,bpcp,cpap,C1}) \/ ({cin,dp} \ {A1,apbp0,apbp,
  bpcp,cpap,C1}) by XBOOLE_1:42
    .= ({cin} \/ {dp}) \ {A1,apbp0,apbp,bpcp,cpap,C1} by A4,ENUMSET1:1
    .= ({cin} \ {A1,apbp0,apbp,bpcp,cpap,C1}) \/ ({dp} \ {A1,apbp0,apbp,bpcp
  ,cpap,C1}) by XBOOLE_1:42
    .= ({cin}) \/ ({dp} \ {A1,apbp0,apbp,bpcp,cpap,C1}) by A2,A5,ZFMISC_1:59
    .= ({cin}) \/ ({dp}) by A3,ZFMISC_1:59
    .= {cin,dp} by ENUMSET1:1;
A7: A1 <> cindp by Lm1;
  InnerVertices S2 misses InputVertices S1 & S1 tolerates S2 by Lm2,CIRCCOMB:47
;
  hence InputVertices (S) = (InputVertices S1) \/ (InputVertices S2 \
  InnerVertices S1) by FACIRC_1:4
    .= ({ap,bp,cp}) \/ (InputVertices S2 \ InnerVertices S1) by GFACIRC1:34
    .= ({ap,bp,cp}) \/ ({A1,cin,dp} \ InnerVertices S1) by A1,A7,GFACIRC1:33
    .= {ap,bp,cp,dp,cin} by A6,ENUMSET1:9;
end;
