A1: ex h being ManySortedSet of NAT st
( 0 -BitMajorityOutput (x,y) = h . 0 & h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)] & ( for n being Nat
for z being set st z = h . n holds
h . (n + 1) = MajorityOutput ((x . (n + 1)),(y . (n + 1)),z) ) ) by Def5;
defpred S₁[ Nat] means n -BitMajorityOutput (x,y) is pair ;
A2: S₁[ 0 ] by A1;
A3: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
(n + 1) -BitMajorityOutput (x,y) = MajorityOutput ((x . (n + 1)),(y . (n + 1)),(n -BitMajorityOutput (x,y))) by Th12
.= [<*[<*(x . (n + 1)),(y . (n + 1))*>,'&'],[<*(y . (n + 1)),(n -BitMajorityOutput (x,y))*>,'&'],[<*(n -BitMajorityOutput (x,y)),(x . (n + 1))*>,'&']*>,or3] ;
hence ( S₁[n] implies S₁[n + 1] ) ; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A2, A3);
hence n -BitMajorityOutput (x,y) is pair ; :: thesis: verum