:: GOBOARD3 semantic presentation
E1:
now
let c
1 be
FinSequence of
(TOP-REAL 2);
let c
2 be
Nat;
assume E2:
len c
1 = c
2 + 1
;
assume
c
2 <> 0
;
then E3:
( 0
< c
2 & c
2 <= c
2 + 1 )
by NAT_1:29;
then
( 0
+ 1
<= c
2 & c
2 <= len c
1 & c
2 + 1
<= len c
1 )
by E2, NAT_1:38;
then E4:
c
2 in dom c
1
by FINSEQ_3:27;
E5:
len (c1 | c2) = c
2
by E2, E3, FINSEQ_1:80;
E6:
dom (c1 | c2) = Seg (len (c1 | c2))
by FINSEQ_1:def 3;
assume E7:
c
1 is
unfolded
;
thus
c
1 | c
2 is
unfolded
proof
set c
3 = c
1 | c
2;
let c
4 be
Nat;
:: according to TOPREAL1:def 8
assume E8:
( 1
<= c
4 & c
4 + 2
<= len (c1 | c2) )
;
then
( c
4 in dom (c1 | c2) & c
4 + 1
in dom (c1 | c2) & c
4 + 2
in dom (c1 | c2) &
(c4 + 1) + 1
= c
4 + (1 + 1) )
by GOBOARD2:4;
then E9:
(
LSeg (c1 | c2),c
4 = LSeg c
1,c
4 &
LSeg (c1 | c2),
(c4 + 1) = LSeg c
1,
(c4 + 1) &
(c1 | c2) /. (c4 + 1) = c
1 /. (c4 + 1) )
by E4, E5, E6, FINSEQ_4:86, TOPREAL3:24;
len (c1 | c2) <= len c
1
by E2, E3, FINSEQ_1:80;
then
c
4 + 2
<= len c
1
by E8, XXREAL_0:2;
hence
(LSeg (c1 | c2),c4) /\ (LSeg (c1 | c2),(c4 + 1)) = {((c1 | c2) /. (c4 + 1))}
by E7, E8, E9, TOPREAL1:def 8;
end;
end;
theorem Th1: :: GOBOARD3:1
theorem Th2: :: GOBOARD3:2