set f1 = and2c ;
set f2 = and2a ;
set f3 = and2 ;
set f0 = xor2c ;
let f, g be nonpair-yielding FinSequence; :: thesis:
deffunc H₁( Nat) -> non empty non void strict unsplit gate`1=arity gate`2isBoolean ManySortedSign = $1 -BitGFA1Str (f,g);
deffunc H₂( set , Nat) -> ManySortedSign = BitGFA1Str ((f . ($2 + 1)),(g . ($2 + 1)),$1);
deffunc H₃( Nat) -> Element of InnerVertices ($1 -BitGFA1Str (f,g)) = $1 -BitGFA1CarryOutput (f,g);
consider h being ManySortedSet of NAT such that
A1: for n being Element of NAT holds h . n = H₃(n) from PBOOLE:sch 5();
deffunc H₄( Nat) -> set = h . $1;
deffunc H₅( set , Nat) -> Element of InnerVertices (GFA1CarryStr ((f . ($2 + 1)),(g . ($2 + 1)),$1)) = GFA1CarryOutput ((f . ($2 + 1)),(g . ($2 + 1)),$1);
set k = (0 -tuples_on BOOLEAN) --> TRUE;
A2: H₁( 0 ) = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) by Th16;
then A3: InnerVertices H₁( 0 ) is Relation by FACIRC_1:38;
A4: InputVertices H₁( 0 ) is without_pairs by A2, FACIRC_1:39;
H₄( 0 ) = H₃( 0 ) by A1;
then A5: h . 0 in InnerVertices H₁( 0 ) ;
A6: for n being Nat
for x being set holds InnerVertices H₂(x,n) is Relation by GFACIRC1:64;

A7: now :: thesis:

let n be Nat; :: thesis:
let x be set ; :: thesis:
assume A8: x = H₄(n) ; :: thesis:
n in NAT by ORDINAL1:def 12;
then A9: H₄(n) = H₃(n) by A1;
then A10: x <> [<*(f . (n + 1)),(g . (n + 1))*>,and2c] by A8, Lm4;
x <> [<*(f . (n + 1)),(g . (n + 1))*>,xor2c] by A8, A9, Lm4;
hence InputVertices H₂(x,n) = {(f . (n + 1)),(g . (n + 1)),x} by A10, GFACIRC1:65; :: thesis:

end;

A11: for n being Nat
for x being set st x = h . n holds
(InputVertices H₂(x,n)) \ {x} is without_pairs

let n be Nat; :: thesis:
let x be set ; :: thesis:
assume x = H₄(n) ; :: thesis:
then A12: InputVertices H₂(x,n) = {(f . (n + 1)),(g . (n + 1)),x} by A7;
thus (InputVertices H₂(x,n)) \ {x} is without_pairs :: thesis:

let a be pair object ; :: according to FACIRC_1:def 2 :: thesis:
assume A13: a in (InputVertices H₂(x,n)) \ {x} ; :: thesis:
then a in InputVertices H₂(x,n) by XBOOLE_0:def 5;
then A14: ( a = f . (n + 1) or a = g . (n + 1) or a = x ) by A12, ENUMSET1:def 1;
not a in {x} by A13, XBOOLE_0:def 5;
hence contradiction by A14, TARSKI:def 1; :: thesis:

end;

end;

A15: now :: thesis:

let n be Nat; :: thesis:
let S be non empty ManySortedSign ; :: thesis:
let x be set ; :: thesis:
assume that
A16: S = H₁(n) and
A17: x = h . n ; :: thesis:
n in NAT by ORDINAL1:def 12;
then A18: x = n -BitGFA1CarryOutput (f,g) by A1, A17;
A19: H₄(n + 1) = (n + 1) -BitGFA1CarryOutput (f,g) by A1;
thus H₁(n + 1) = S +* H₂(x,n) by A16, A18, Th21; :: thesis:
thus h . (n + 1) = H₅(x,n) by A18, A19, Th21; :: thesis:
InputVertices H₂(x,n) = {(f . (n + 1)),(g . (n + 1)),x} by A7, A17;
hence x in InputVertices H₂(x,n) by ENUMSET1:def 1; :: thesis:
A20: InnerVertices H₂(x,n) = (({[<*(f . (n + 1)),(g . (n + 1))*>,xor2c]} \/ {(GFA1AdderOutput ((f . (n + 1)),(g . (n + 1)),x))}) \/ {[<*(f . (n + 1)),(g . (n + 1))*>,and2c],[<*(g . (n + 1)),x*>,and2a],[<*x,(f . (n + 1))*>,and2]}) \/ {(GFA1CarryOutput ((f . (n + 1)),(g . (n + 1)),x))} by GFACIRC1:63;
H₅(x,n) in {H₅(x,n)} by TARSKI:def 1;
hence H₅(x,n) in InnerVertices H₂(x,n) by A20, XBOOLE_0:def 3; :: thesis:

end;

A21: for n being Nat holds
( InputVertices H₁(n + 1) = (InputVertices H₁(n)) \/ ((InputVertices H₂(h . n,n)) \ {(h . n)}) & InnerVertices H₁(n) is Relation & InputVertices H₁(n) is without_pairs ) from CIRCCMB2:sch 11(A3, A4, A5, A6, A11, A15);
let n be Nat; :: thesis:
n in NAT by ORDINAL1:def 12;
then h . n = n -BitGFA1CarryOutput (f,g) by A1;
hence ( InputVertices ((n + 1) -BitGFA1Str (f,g)) = (InputVertices (n -BitGFA1Str (f,g))) \/ ((InputVertices (BitGFA1Str ((f . (n + 1)),(g . (n + 1)),(n -BitGFA1CarryOutput (f,g))))) \ {(n -BitGFA1CarryOutput (f,g))}) & InnerVertices (n -BitGFA1Str (f,g)) is Relation & InputVertices (n -BitGFA1Str (f,g)) is without_pairs ) by A21; :: thesis: