let a, b be FinSequence; ( 0 -BitAdderStr (a,b) = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) & 0 -BitAdderCirc (a,b) = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) & 0 -BitMajorityOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)] )
A1:
ex f, g, h being ManySortedSet of NAT st
( 0 -BitAdderStr (a,b) = f . 0 & 0 -BitAdderCirc (a,b) = g . 0 & f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) & g . 0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) & h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)] & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for z being set st S = f . n & A = g . n & z = h . n holds
( f . (n + 1) = S +* (BitAdderWithOverflowStr ((a . (n + 1)),(b . (n + 1)),z)) & g . (n + 1) = A +* (BitAdderWithOverflowCirc ((a . (n + 1)),(b . (n + 1)),z)) & h . (n + 1) = MajorityOutput ((a . (n + 1)),(b . (n + 1)),z) ) ) )
by Def4;
hence
0 -BitAdderStr (a,b) = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> FALSE))
; ( 0 -BitAdderCirc (a,b) = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) & 0 -BitMajorityOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)] )
thus
0 -BitAdderCirc (a,b) = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> FALSE))
by A1; 0 -BitMajorityOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)]
InnerVertices (0 -BitAdderStr (a,b)) = {[<*>,((0 -tuples_on BOOLEAN) --> FALSE)]}
by A1, CIRCCOMB:42;
hence
0 -BitMajorityOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)]
by TARSKI:def 1; verum