reconsider n = n as Nat ;
set S0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE );
set o0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )];
deffunc H₁( non empty ManySortedSign , set , Nat) -> ManySortedSign = $1 +* (BitGFA1Str (x . ($3 + 1)),(y . ($3 + 1)),$2);
deffunc H₂( set , Nat) -> Element of InnerVertices (GFA1CarryStr (x . ($2 + 1)),(y . ($2 + 1)),$1) = GFA1CarryOutput (x . ($2 + 1)),(y . ($2 + 1)),$1;
for S1, S2 being non empty non void strict unsplit gate`1=arity gate`2isBoolean ManySortedSign st ex f, h being ManySortedSet of st
( S1 = f . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] & ( 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) = H₁(S,x,n) & h . (n + 1) = H₂(x,n) ) ) ) & ex f, h being ManySortedSet of st
( S2 = f . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] & ( 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) = H₁(S,x,n) & h . (n + 1) = H₂(x,n) ) ) ) holds
S1 = S2 from CIRCCMB2:sch 9();
hence for b₁, b₂ being non empty non void strict unsplit gate`1=arity gate`2isBoolean ManySortedSign st ex f, h being ManySortedSet of st
( b₁ = f . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] & ( 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 +* (BitGFA1Str (x . (n + 1)),(y . (n + 1)),z) & h . (n + 1) = GFA1CarryOutput (x . (n + 1)),(y . (n + 1)),z ) ) ) & ex f, h being ManySortedSet of st
( b₂ = f . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] & ( 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 +* (BitGFA1Str (x . (n + 1)),(y . (n + 1)),z) & h . (n + 1) = GFA1CarryOutput (x . (n + 1)),(y . (n + 1)),z ) ) ) holds
b₁ = b₂ ; :: thesis: verum