let f, g be nonpair-yielding FinSequence; :: thesis:
deffunc H₁( Nat) -> non empty non void strict unsplit gate`1=arity gate`2isBoolean ManySortedSign = $1 -BitAdderStr (f,g);
deffunc H₂( set , Nat) -> ManySortedSign = BitAdderWithOverflowStr ((f . ($2 + 1)),(g . ($2 + 1)),$1);
deffunc H₃( Nat) -> Element of InnerVertices ($1 -BitAdderStr (f,g)) = $1 -BitMajorityOutput (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();
A2: for n being Nat holds h . n = H₃(n)

let n be Nat; :: thesis:
n in NAT by ORDINAL1:def 12;
hence h . n = H₃(n) by A1; :: thesis:

end;

deffunc H₄( Nat) -> set = h . $1;
deffunc H₅( set , Nat) -> Element of InnerVertices (MajorityStr ((f . ($2 + 1)),(g . ($2 + 1)),$1)) = MajorityOutput ((f . ($2 + 1)),(g . ($2 + 1)),$1);
set k = (0 -tuples_on BOOLEAN) --> FALSE;
A3: 0 -BitAdderStr (f,g) = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) by Th7;
then A4: InnerVertices H₁( 0 ) is Relation by FACIRC_1:38;
A5: InputVertices H₁( 0 ) is without_pairs by A3, FACIRC_1:39;
H₄( 0 ) = 0 -BitMajorityOutput (f,g) by A2;
then A6: h . 0 in InnerVertices H₁( 0 ) ;
A7: for n being Nat
for x being set holds InnerVertices H₂(x,n) is Relation by FACIRC_1:88;

A8: now :: thesis:

let n be Nat; :: thesis:
let x be set ; :: thesis:
assume A9: x = H₄(n) ; :: thesis:
A10: H₄(n) = n -BitMajorityOutput (f,g) by A2;
then A11: x <> [<*(f . (n + 1)),(g . (n + 1))*>,'&'] by A9, Th25;
x <> [<*(f . (n + 1)),(g . (n + 1))*>,'xor'] by A9, A10, Th25;
hence InputVertices H₂(x,n) = {(f . (n + 1)),(g . (n + 1)),x} by A11, Th22; :: thesis:

end;

A12: 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 A13: InputVertices H₂(x,n) = {(f . (n + 1)),(g . (n + 1)),x} by A8;
thus (InputVertices H₂(x,n)) \ {x} is without_pairs :: thesis:

let a be pair object ; :: according to FACIRC_1:def 2 :: thesis:
assume A14: a in (InputVertices H₂(x,n)) \ {x} ; :: thesis:
then a in InputVertices H₂(x,n) by XBOOLE_0:def 5;
then A15: ( a = f . (n + 1) or a = g . (n + 1) or a = x ) by A13, ENUMSET1:def 1;
not a in {x} by A14, XBOOLE_0:def 5;
hence contradiction by A15, TARSKI:def 1; :: thesis:

end;

end;

A16: now :: thesis:

let n be Nat; :: thesis:
let S be non empty ManySortedSign ; :: thesis:
let x be set ; :: thesis:
assume that
A17: S = H₁(n) and
A18: x = h . n ; :: thesis:
A19: x = n -BitMajorityOutput (f,g) by A2, A18;
A20: H₄(n + 1) = (n + 1) -BitMajorityOutput (f,g) by A2;
thus H₁(n + 1) = S +* H₂(x,n) by A17, A19, Th12; :: thesis:
thus h . (n + 1) = H₅(x,n) by A19, A20, Th12; :: thesis:
InputVertices H₂(x,n) = {(f . (n + 1)),(g . (n + 1)),x} by A8, A18;
hence x in InputVertices H₂(x,n) by ENUMSET1:def 1; :: thesis:
A21: InnerVertices H₂(x,n) = ({[<*(f . (n + 1)),(g . (n + 1))*>,'xor'],(2GatesCircOutput ((f . (n + 1)),(g . (n + 1)),x,'xor'))} \/ {[<*(f . (n + 1)),(g . (n + 1))*>,'&'],[<*(g . (n + 1)),x*>,'&'],[<*x,(f . (n + 1))*>,'&']}) \/ {(MajorityOutput ((f . (n + 1)),(g . (n + 1)),x))} by Th23;
H₅(x,n) in {H₅(x,n)} by TARSKI:def 1;
hence H₅(x,n) in InnerVertices H₂(x,n) by A21, XBOOLE_0:def 3; :: thesis:

end;

A22: 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(A4, A5, A6, A7, A12, A16);
let n be Nat; :: thesis:
h . n = n -BitMajorityOutput (f,g) by A2;
hence ( InputVertices ((n + 1) -BitAdderStr (f,g)) = (InputVertices (n -BitAdderStr (f,g))) \/ ((InputVertices (BitAdderWithOverflowStr ((f . (n + 1)),(g . (n + 1)),(n -BitMajorityOutput (f,g))))) \ {(n -BitMajorityOutput (f,g))}) & InnerVertices (n -BitAdderStr (f,g)) is Relation & InputVertices (n -BitAdderStr (f,g)) is without_pairs ) by A22; :: thesis: