
theorem Th11:
  for ap,bm,cp,dm,cin being set holds InnerVertices BitFTA1Str(ap,
bm,cp,dm,cin) = {[<*ap,bm*>,xor2c], GFA1AdderOutput(ap,bm,cp)} \/ {[<*ap,bm*>,
and2c], [<*bm,cp*>,and2a], [<*cp,ap*>,and2], GFA1CarryOutput(ap,bm,cp)} \/ {[<*
GFA1AdderOutput(ap,bm,cp),cin*>,xor2c], GFA2AdderOutput(GFA1AdderOutput(ap,bm,
cp),cin,dm)} \/ {[<*GFA1AdderOutput(ap,bm,cp),cin*>,and2a], [<*cin,dm*>,and2c],
[<*dm,GFA1AdderOutput(ap,bm,cp)*>,nor2], GFA2CarryOutput(GFA1AdderOutput(ap,bm
  ,cp),cin,dm)}
proof
  let ap,bm,cp,dm,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 A2 = GFA2AdderOutput(A1,cin,dm);
  set C2 = GFA2CarryOutput(A1,cin,dm);
  set apbm0 = [<*ap,bm*>, xor2c];
  set apbm = [<*ap,bm*>, and2c];
  set bmcp = [<*bm,cp*>, and2a];
  set cpap = [<*cp,ap*>, and2 ];
  set A1cin0 = [<*A1,cin*>,xor2c];
  set A1cin = [<*A1,cin*>,and2a];
  set cindm = [<*cin,dm*>,and2c];
  set dmA1 = [<*dm,A1*>, nor2];
  S1 tolerates S2 by CIRCCOMB:47;
  hence InnerVertices S = (InnerVertices S1) \/ (InnerVertices S2) by
CIRCCOMB:11
    .= ({apbm0} \/ {A1} \/ {apbm,bmcp,cpap} \/ {C1}) \/ (InnerVertices S2)
  by GFACIRC1:63
    .= ({apbm0,A1} \/ {apbm,bmcp,cpap} \/ {C1}) \/ (InnerVertices S2) by
ENUMSET1:1
    .= ({apbm0,A1} \/ ({apbm,bmcp,cpap} \/ {C1})) \/ (InnerVertices S2) by
XBOOLE_1:4
    .= ({apbm0,A1} \/ {apbm,bmcp,cpap,C1}) \/ (InnerVertices S2) by ENUMSET1:6
    .= ({apbm0,A1} \/ {apbm,bmcp,cpap,C1}) \/ ({A1cin0} \/ {A2} \/ {A1cin,
  cindm,dmA1} \/ {C2}) by GFACIRC1:95
    .= ({apbm0,A1} \/ {apbm,bmcp,cpap,C1}) \/ ({A1cin0,A2} \/ {A1cin,cindm,
  dmA1} \/ {C2}) by ENUMSET1:1
    .= ({apbm0,A1} \/ {apbm,bmcp,cpap,C1}) \/ ({A1cin0,A2} \/ ({A1cin,cindm,
  dmA1} \/ {C2})) by XBOOLE_1:4
    .= ({apbm0,A1} \/ {apbm,bmcp,cpap,C1}) \/ ({A1cin0,A2} \/ {A1cin,cindm,
  dmA1,C2}) by ENUMSET1:6
    .= {apbm0,A1} \/ {apbm,bmcp,cpap,C1} \/ {A1cin0,A2} \/ {A1cin,cindm,dmA1
  ,C2} by XBOOLE_1:4;
end;
