let a, b be FinSequence; ( 0 -BitGFA1Str (a,b) = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & 0 -BitGFA1Circ (a,b) = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & 0 -BitGFA1CarryOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] )
set f0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE));
set g0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE));
set h0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)];
A1:
ex f, g, h being ManySortedSet of NAT st
( 0 -BitGFA1Str (a,b) = f . 0 & 0 -BitGFA1Circ (a,b) = g . 0 & f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & g . 0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] & ( 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 +* (BitGFA1Str ((a . (n + 1)),(b . (n + 1)),z)) & g . (n + 1) = A +* (BitGFA1Circ ((a . (n + 1)),(b . (n + 1)),z)) & h . (n + 1) = GFA1CarryOutput ((a . (n + 1)),(b . (n + 1)),z) ) ) )
by Def6;
hence
0 -BitGFA1Str (a,b) = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE))
; ( 0 -BitGFA1Circ (a,b) = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & 0 -BitGFA1CarryOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] )
thus
0 -BitGFA1Circ (a,b) = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE))
by A1; 0 -BitGFA1CarryOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)]
InnerVertices (0 -BitGFA1Str (a,b)) = {[<*>,((0 -tuples_on BOOLEAN) --> TRUE)]}
by A1, CIRCCOMB:42;
hence
0 -BitGFA1CarryOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)]
by TARSKI:def 1; verum