let f be s.c.c. FinSequence of (TOP-REAL 2); :: thesis: for n being Nat st n < len f holds
f | n is s.n.c.
let n be 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 7 :: 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_5:17;

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 3;
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 3;
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 A4: j + 1 <= n and
A5: j + 1 <= len (f | n) ; :: thesis: LSeg ((f | n),i) misses LSeg ((f | n),j)

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

end;

end;