A1:
ex h being ManySortedSet of NAT st
( 0 -BitBorrowOutput (x,y) = h . 0 & h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] & ( for n being Nat holds h . (n + 1) = BorrowOutput ((x . (n + 1)),(y . (n + 1)),(h . n)) ) )
by Def3;
defpred S1[ Nat] means n -BitBorrowOutput (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;
( S1[n] implies S1[n + 1] )
(n + 1) -BitBorrowOutput (
x,
y) =
BorrowOutput (
(x . (n + 1)),
(y . (n + 1)),
(n -BitBorrowOutput (x,y)))
by Th7
.=
[<*[<*(x . (n + 1)),(y . (n + 1))*>,and2a],[<*(y . (n + 1)),(n -BitBorrowOutput (x,y))*>,and2],[<*(x . (n + 1)),(n -BitBorrowOutput (x,y))*>,and2a]*>,or3]
;
hence
(
S1[
n] implies
S1[
n + 1] )
;
verum
end;
for n being Nat holds S1[n]
from NAT_1:sch 2(A2, A3);
hence
n -BitBorrowOutput (x,y) is pair
; verum