reconsider n = n as Nat ;
deffunc H₁( set , Nat) -> ManySortedSign = BitGFA0Str (x . ($2 + 1)),(y . ($2 + 1)),$1;
deffunc H₂( set , Nat) -> Element of InnerVertices (GFA0CarryStr (x . ($2 + 1)),(y . ($2 + 1)),$1) = GFA0CarryOutput (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 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 +* H₁(x,n) & h . (n + 1) = H₂(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, h being ManySortedSet of st
( b₁ = 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 z being set st S = f . n & z = h . n holds
( f . (n + 1) = S +* (BitGFA0Str (x . (n + 1)),(y . (n + 1)),z) & h . (n + 1) = GFA0CarryOutput (x . (n + 1)),(y . (n + 1)),z ) ) ) ; :: thesis: verum