
theorem
  for am,bm,cm being non pair set for dm,cin being set for s being State
of BitFTA3Circ(am,bm,cm,dm,cin) for a1,a2,a3 being Element of BOOLEAN st a1 = s
.am & a2 = s.bm & a3 = s.cm holds Following(s,2).BitFTA3CarryOutput(am,bm,cm,dm
,cin) = 'not' (('not' a1 '&' 'not' a2) 'or' ('not' a2 '&' 'not' a3) 'or' ('not'
a3 '&' 'not' a1)) & Following(s,2).BitFTA3AdderOutputI(am,bm,cm,dm,cin) = 'not'
  ('not' a1 'xor' 'not' a2 'xor' 'not' a3)
proof
  let am,bm,cm be non pair set;
  let dm,cin be set;
  let s be State of BitFTA3Circ(am,bm,cm,dm,cin);
  set S1 = BitGFA3Str(am,bm,cm);
  set C1 = BitGFA3Circ(am,bm,cm);
  set A1 = GFA3AdderOutput(am,bm,cm);
  set A2 = GFA3CarryOutput(am,bm,cm);
  set S2 = BitGFA3Str(A1,cin,dm);
  set C2 = BitGFA3Circ(A1,cin,dm);
  let a1,a2,a3 be Element of BOOLEAN such that
A1: a1 = s.am and
A2: a2 = s.bm and
A3: a3 = s.cm;
  reconsider s1 = s|the carrier of S1 as State of C1 by FACIRC_1:26;
A4: dom s1 = the carrier of S1 by CIRCUIT1:3;
  am in the carrier of S1 by GFACIRC1:132;
  then
A5: a1 = s1.am by A1,A4,FUNCT_1:47;
  reconsider t = s as State of C1+*C2;
A6: InputVertices S1 misses InnerVertices S2 by Lm32;
  cm in the carrier of S1 by GFACIRC1:132;
  then
A7: a3 = s1.cm by A3,A4,FUNCT_1:47;
  bm in the carrier of S1 by GFACIRC1:132;
  then
A8: a2 = s1.bm by A2,A4,FUNCT_1:47;
  A2 in the carrier of S1 by GFACIRC1:132;
  then Following(t,2).GFA3CarryOutput(am,bm,cm) = Following(s1,2).
  GFA3CarryOutput(am,bm,cm) by A6,FACIRC_1:32;
  hence Following(s,2).BitFTA3CarryOutput(am,bm,cm,dm,cin) = 'not' (('not' a1
'&' 'not' a2) 'or' ('not' a2 '&' 'not' a3) 'or' ('not' a3 '&' 'not' a1)) by A5
,A8,A7,GFACIRC1:135;
  A1 in the carrier of S1 by GFACIRC1:132;
  then Following(t,2).GFA3AdderOutput(am,bm,cm) = Following(s1,2).
  GFA3AdderOutput(am,bm,cm) by A6,FACIRC_1:32;
  hence thesis by A5,A8,A7,GFACIRC1:135;
end;
