let x, y be FinSequence; for n being Nat holds
( ( (n -BitBorrowOutput (x,y)) `1 = <*> & (n -BitBorrowOutput (x,y)) `2 = (0 -tuples_on BOOLEAN) --> TRUE & proj1 ((n -BitBorrowOutput (x,y)) `2) = 0 -tuples_on BOOLEAN ) or ( card ((n -BitBorrowOutput (x,y)) `1) = 3 & (n -BitBorrowOutput (x,y)) `2 = or3 & proj1 ((n -BitBorrowOutput (x,y)) `2) = 3 -tuples_on BOOLEAN ) )
defpred S1[ Nat] means ( ( ($1 -BitBorrowOutput (x,y)) `1 = <*> & ($1 -BitBorrowOutput (x,y)) `2 = (0 -tuples_on BOOLEAN) --> TRUE & proj1 (($1 -BitBorrowOutput (x,y)) `2) = 0 -tuples_on BOOLEAN ) or ( card (($1 -BitBorrowOutput (x,y)) `1) = 3 & ($1 -BitBorrowOutput (x,y)) `2 = or3 & proj1 (($1 -BitBorrowOutput (x,y)) `2) = 3 -tuples_on BOOLEAN ) );
A1:
dom ((0 -tuples_on BOOLEAN) --> TRUE) = 0 -tuples_on BOOLEAN
by FUNCOP_1:13;
0 -BitBorrowOutput (x,y) = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)]
by Th2;
then A2:
S1[ 0 ]
by A1;
A3:
now for n being Nat st S1[n] holds
S1[n + 1]let n be
Nat;
( S1[n] implies S1[n + 1] )assume
S1[
n]
;
S1[n + 1]set c =
n -BitBorrowOutput (
x,
y);
A4:
(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]
;
A5:
dom or3 = 3
-tuples_on BOOLEAN
by FUNCT_2:def 1;
thus
S1[
n + 1]
by A4, A5, FINSEQ_1:45;
verum end;
thus
for i being Nat holds S1[i]
from NAT_1:sch 2(A2, A3); verum