reconsider n = n as Element of NAT by ORDINAL1:def 13;
deffunc H₁( set , Nat) -> Element of InnerVertices (MajorityStr (x . ($2 + 1)),(y . ($2 + 1)),$1) = MajorityOutput (x . ($2 + 1)),(y . ($2 + 1)),$1;
deffunc H₂( set , Nat) -> ManySortedSign = BitAdderWithOverflowStr (x . ($2 + 1)),(y . ($2 + 1)),$1;
ex S being non empty non void strict unsplit gate`1=arity gate`2isBoolean ManySortedSign ex f, h being ManySortedSet of NAT st
( S = f . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )] & ( for n being Nat
for S being non empty ManySortedSign
for x being set st S = f . n & x = h . n holds
( f . (n + 1) = S +* H₂(x,n) & h . (n + 1) = H₁(x,n) ) ) ) from CIRCCMB2:sch 8(A1);
hence ex b₁ being non empty non void strict unsplit gate`1=arity gate`2isBoolean ManySortedSign ex f, g being ManySortedSet of NAT st
( b₁ = f . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) & g . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )] & ( for n being Nat
for S being non empty ManySortedSign
for z being set st S = f . n & z = g . n holds
( f . (n + 1) = S +* (BitAdderWithOverflowStr (x . (n + 1)),(y . (n + 1)),z) & g . (n + 1) = MajorityOutput (x . (n + 1)),(y . (n + 1)),z ) ) ) ; :: thesis: verum