reconsider n = n as Element of NAT by ORDINAL1:def 12;
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) ) ) )
; verum