:: GOBOARD3 semantic presentation
E36:
now
let f be
FinSequence of
(TOP-REAL 2);
let k be
Element of
NAT ;
assume E37:
len f = k + 1
;
assume
k <> 0
;
then E38:
( 0
< k &
k <= k + 1 )
by NAT_1:29;
then
( 0
+ 1
<= k &
k <= len f &
k + 1
<= len f )
by , NAT_1:38;
then E39:
k in dom f
by FINSEQ_3:27;
E40:
len (f | k) = k
by , , FINSEQ_1:80;
E41:
dom (f | k) = Seg (len (f | k))
by FINSEQ_1:def 3;
assume E42:
f is
unfolded
;
thus
f | k is
unfolded
proof
set f1 =
f | k;
let n be
Element of
NAT ;
:: according to TOPREAL1:def 8
assume E44:
( 1
<= n &
n + 2
<= len (f | k) )
;
then
(
n in dom (f | k) &
n + 1
in dom (f | k) &
n + 2
in dom (f | k) &
(n + 1) + 1
= n + (1 + 1) )
by GOBOARD2:4;
then E45:
(
LSeg (f | k),
n = LSeg f,
n &
LSeg (f | k),
(n + 1) = LSeg f,
(n + 1) &
(f | k) /. (n + 1) = f /. (n + 1) )
by , , , FINSEQ_4:86, TOPREAL3:24;
len (f | k) <= len f
by , , FINSEQ_1:80;
then
n + 2
<= len f
by , XXREAL_0:2;
hence
(LSeg (f | k),n) /\ (LSeg (f | k),(n + 1)) = {((f | k) /. (n + 1))}
by , , , TOPREAL1:def 8;
end;
end;
theorem Th1: :: GOBOARD3:1
theorem Th2: :: GOBOARD3:2