
theorem Th21:
  for am,bp,cm,dp,cin being set holds InnerVertices BitFTA2Str(am,
bp,cm,dp,cin) = {[<*am,bp*>,xor2c], GFA2AdderOutput(am,bp,cm)} \/ {[<*am,bp*>,
and2a], [<*bp,cm*>,and2c], [<*cm,am*>,nor2], GFA2CarryOutput(am,bp,cm)} \/ {[
<*GFA2AdderOutput(am,bp,cm),cin*>,xor2c], GFA1AdderOutput(GFA2AdderOutput(am,bp
,cm),cin,dp)} \/ {[<*GFA2AdderOutput(am,bp,cm),cin*>,and2c], [<*cin,dp*>,and2a]
, [<*dp,GFA2AdderOutput(am,bp,cm)*>,and2], GFA1CarryOutput(GFA2AdderOutput(am,
  bp,cm),cin,dp)}
proof
  let am,bp,cm,dp,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 A2 = GFA1AdderOutput(A1,cin,dp);
  set C2 = GFA1CarryOutput(A1,cin,dp);
  set ambp0 = [<*am,bp*>, xor2c];
  set ambp = [<*am,bp*>, and2a];
  set bpcm = [<*bp,cm*>, and2c];
  set cmam = [<*cm,am*>, nor2];
  set A1cin0 = [<*A1,cin*>,xor2c];
  set A1cin = [<*A1,cin*>,and2c];
  set cindp = [<*cin,dp*>,and2a];
  set dpA1 = [<*dp,A1*>, and2 ];
  S1 tolerates S2 by CIRCCOMB:47;
  hence InnerVertices S = (InnerVertices S1) \/ (InnerVertices S2) by
CIRCCOMB:11
    .= ({ambp0} \/ {A1} \/ {ambp,bpcm,cmam} \/ {C1}) \/ (InnerVertices S2)
  by GFACIRC1:95
    .= ({ambp0,A1} \/ {ambp,bpcm,cmam} \/ {C1}) \/ (InnerVertices S2) by
ENUMSET1:1
    .= ({ambp0,A1} \/ ({ambp,bpcm,cmam} \/ {C1})) \/ (InnerVertices S2) by
XBOOLE_1:4
    .= ({ambp0,A1} \/ {ambp,bpcm,cmam,C1}) \/ (InnerVertices S2) by ENUMSET1:6
    .= ({ambp0,A1} \/ {ambp,bpcm,cmam,C1}) \/ ({A1cin0} \/ {A2} \/ {A1cin,
  cindp,dpA1} \/ {C2}) by GFACIRC1:63
    .= ({ambp0,A1} \/ {ambp,bpcm,cmam,C1}) \/ ({A1cin0,A2} \/ {A1cin,cindp,
  dpA1} \/ {C2}) by ENUMSET1:1
    .= ({ambp0,A1} \/ {ambp,bpcm,cmam,C1}) \/ ({A1cin0,A2} \/ ({A1cin,cindp,
  dpA1} \/ {C2})) by XBOOLE_1:4
    .= ({ambp0,A1} \/ {ambp,bpcm,cmam,C1}) \/ ({A1cin0,A2} \/ {A1cin,cindp,
  dpA1,C2}) by ENUMSET1:6
    .= {ambp0,A1} \/ {ambp,bpcm,cmam,C1} \/ {A1cin0,A2} \/ {A1cin,cindp,dpA1
  ,C2} by XBOOLE_1:4;
end;
