let f be s.c.c. FinSequence of (TOP-REAL 2); :: thesis: for n being Element of NAT st n < len f holds
f | n is s.n.c.
let n be Element of NAT ; :: thesis: ( n < len f implies f | n is s.n.c. )
assume A1: n < len f ; :: thesis: f | n is s.n.c.
let i, j be Nat; :: according to TOPREAL1:def 9 :: thesis: ( j <= i + 1 or LSeg (f | n),i misses LSeg (f | n),j )
assume A2: i + 1 < j ; :: thesis: LSeg (f | n),i misses LSeg (f | n),j
A3: len (f | n) <= n by FINSEQ_5:19;
A4: ( i in NAT & j in NAT ) by ORDINAL1:def 13;

per cases ( n < j + 1 or len (f | n) < j + 1 or ( j + 1 <= n & j + 1 <= len (f | n) ) ) ;

suppose n < j + 1 ; :: thesis: LSeg (f | n),i misses LSeg (f | n),j

then len (f | n) < j + 1 by A3, XXREAL_0:2;
then LSeg (f | n),j = {} by TOPREAL1:def 5;
then (LSeg (f | n),i) /\ (LSeg (f | n),j) = {} ;
hence LSeg (f | n),i misses LSeg (f | n),j by XBOOLE_0:def 7; :: thesis: verum

end;

suppose len (f | n) < j + 1 ; :: thesis: LSeg (f | n),i misses LSeg (f | n),j

then LSeg (f | n),j = {} by TOPREAL1:def 5;
then (LSeg (f | n),i) /\ (LSeg (f | n),j) = {} ;
hence LSeg (f | n),i misses LSeg (f | n),j by XBOOLE_0:def 7; :: thesis: verum

end;

suppose that A5: j + 1 <= n and
A6: j + 1 <= len (f | n) ; :: thesis: LSeg (f | n),i misses LSeg (f | n),j

j + 1 < len f by A1, A5, XXREAL_0:2;
then A7: LSeg f,i misses LSeg f,j by A2, A4, GOBOARD5:def 4;
A8: LSeg f,j = LSeg (f | n),j by A6, SPPOL_2:3;
j <= j + 1 by NAT_1:11;
then i + 1 <= j + 1 by A2, XXREAL_0:2;
hence LSeg (f | n),i misses LSeg (f | n),j by A6, A7, A8, SPPOL_2:3, XXREAL_0:2; :: thesis: verum

end;

end;