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) = H₂(S,x,n) & h . (n + 1) = H₁(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) = H₂(S,x,n) & h . (n + 1) = H₁(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, g being ManySortedSet of st
( b₁ = f . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & g . 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 = g . n holds
( f . (n + 1) = S +* (BitSubtracterWithBorrowStr (x . (n + 1)),(y . (n + 1)),z) & g . (n + 1) = BorrowOutput (x . (n + 1)),(y . (n + 1)),z ) ) ) & ex f, g being ManySortedSet of st
( b₂ = f . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & g . 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 = g . n holds
( f . (n + 1) = S +* (BitSubtracterWithBorrowStr (x . (n + 1)),(y . (n + 1)),z) & g . (n + 1) = BorrowOutput (x . (n + 1)),(y . (n + 1)),z ) ) ) holds
b₁ = b₂ ; :: thesis: verum