
theorem Th4:
  for ap,bp,cp,dp,cin being set holds ap in the carrier of
BitFTA0Str(ap,bp,cp,dp,cin) & bp in the carrier of BitFTA0Str(ap,bp,cp,dp,cin)
  & cp in the carrier of BitFTA0Str(ap,bp,cp,dp,cin) & dp in the carrier of
BitFTA0Str(ap,bp,cp,dp,cin) & cin in the carrier of BitFTA0Str(ap,bp,cp,dp,cin)
  & [<*ap,bp*>,xor2] in the carrier of BitFTA0Str(ap,bp,cp,dp,cin) &
GFA0AdderOutput(ap,bp,cp) in the carrier of BitFTA0Str(ap,bp,cp,dp,cin) & [<*ap
,bp*>,and2] in the carrier of BitFTA0Str(ap,bp,cp,dp,cin) & [<*bp,cp*>,and2] in
the carrier of BitFTA0Str(ap,bp,cp,dp,cin) & [<*cp,ap*>,and2] in the carrier of
  BitFTA0Str(ap,bp,cp,dp,cin) & GFA0CarryOutput(ap,bp,cp) in the carrier of
  BitFTA0Str(ap,bp,cp,dp,cin) & [<*GFA0AdderOutput(ap,bp,cp),cin*>,xor2] in the
carrier of BitFTA0Str(ap,bp,cp,dp,cin) & GFA0AdderOutput(GFA0AdderOutput(ap,bp,
cp),cin,dp) in the carrier of BitFTA0Str(ap,bp,cp,dp,cin) & [<*GFA0AdderOutput(
ap,bp,cp),cin*>,and2] in the carrier of BitFTA0Str(ap,bp,cp,dp,cin) & [<*cin,dp
*>,and2] in the carrier of BitFTA0Str(ap,bp,cp,dp,cin) & [<*dp,GFA0AdderOutput(
  ap,bp,cp)*>,and2] in the carrier of BitFTA0Str(ap,bp,cp,dp,cin) &
GFA0CarryOutput(GFA0AdderOutput(ap,bp,cp),cin,dp) in the carrier of BitFTA0Str(
  ap,bp,cp,dp,cin)
proof
  let ap,bp,cp,dp,cin be set;
  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 A2 = GFA0AdderOutput(A1,cin,dp);
  set C2 = GFA0CarryOutput(A1,cin,dp);
  set apbp0 = [<*ap,bp*>, xor2];
  set apbp = [<*ap,bp*>, and2];
  set bpcp = [<*bp,cp*>, and2];
  set cpap = [<*cp,ap*>, and2];
  set A1cin0 = [<*A1,cin*>,xor2];
  set A1cin = [<*A1,cin*>,and2];
  set cindp = [<*cin,dp*>,and2];
  set dpA1 = [<*dp,A1*>, and2];
A1: cp in the carrier of S1 & apbp0 in the carrier of S1 by GFACIRC1:36;
A2: apbp in the carrier of S1 & bpcp in the carrier of S1 by GFACIRC1:36;
A3: A1 in the carrier of S2 & cin in the carrier of S2 by GFACIRC1:36;
A4: cpap in the carrier of S1 & C1 in the carrier of S1 by GFACIRC1:36;
A5: C2 in the carrier of S2 by GFACIRC1:36;
A6: cindp in the carrier of S2 & dpA1 in the carrier of S2 by GFACIRC1:36;
A7: A2 in the carrier of S2 & A1cin in the carrier of S2 by GFACIRC1:36;
A8: dp in the carrier of S2 & A1cin0 in the carrier of S2 by GFACIRC1:36;
  ap in the carrier of S1 & bp in the carrier of S1 by GFACIRC1:36;
  hence thesis by A1,A2,A4,A3,A8,A7,A6,A5,FACIRC_1:20;
end;
