let f be FinSequence of (TOP-REAL 2); :: thesis: for n being Element of NAT st f is s.n.c. holds
f /^ n is s.n.c.
let n be Element of NAT ; :: thesis: ( f is s.n.c. implies f /^ n is s.n.c. )
assume A1: f is s.n.c. ; :: thesis: f /^ n is s.n.c.

per cases ( n <= len f or n > len f ) ;

suppose A2: 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 A3: i + 1 < j ; :: thesis: LSeg (f /^ n),i misses LSeg (f /^ n),j

now

per cases ( i = 0 or j + 1 > len (f /^ n) or ( i <> 0 & j + 1 <= len (f /^ n) ) ) ;

suppose A4: ( i = 0 or j + 1 > len (f /^ n) ) ; :: thesis: LSeg (f /^ n),i misses LSeg (f /^ n),j

now

per cases ( i = 0 or j + 1 > len (f /^ n) ) by A4;

case i = 0 ; :: thesis: LSeg (f /^ n),i = {}

hence LSeg (f /^ n),i = {} by TOPREAL1:def 5; :: thesis: verum

end;

case j + 1 > len (f /^ n) ; :: thesis: LSeg (f /^ n),j = {}

hence LSeg (f /^ n),j = {} by TOPREAL1:def 5; :: thesis: verum

end;

end;

end;

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: i <> 0 and
A6: j + 1 <= len (f /^ n) ; :: thesis: LSeg (f /^ n),i misses LSeg (f /^ n),j

A7: len (f /^ n) = (len f) - n by A2, RFINSEQ:def 2;
A8: 1 <= i by A5, NAT_1:14;
i <= j by A3, NAT_1:13;
then A9: 1 <= j by A8, XXREAL_0:2;
n + (i + 1) < n + j by A3, XREAL_1:8;
then (n + i) + 1 < n + j ;
then A10: LSeg f,(n + i) misses LSeg f,(n + j) by A1, TOPREAL1:def 9;
j <= j + 1 by NAT_1:11;
then i + 1 < j + 1 by A3, XXREAL_0:2;
then i + 1 <= len (f /^ n) by A6, XXREAL_0:2;
then (LSeg (f /^ n),i) /\ (LSeg (f /^ n),j) = (LSeg f,(n + i)) /\ (LSeg (f /^ n),j) by A5, A7, Th5, NAT_1:14
.= (LSeg f,(n + i)) /\ (LSeg f,(n + j)) by A6, A7, A9, Th5
.= {} by A10, XBOOLE_0:def 7 ;
hence LSeg (f /^ n),i misses LSeg (f /^ n),j by XBOOLE_0:def 7; :: thesis: verum

end;

end;

end;

hence LSeg (f /^ n),i misses LSeg (f /^ n),j ; :: thesis: verum

end;

suppose n > len f ; :: thesis: f /^ n is s.n.c.

then f /^ n = <*> the carrier of (TOP-REAL 2) by RFINSEQ:def 2;
then len (f /^ n) = 0 ;
hence f /^ n is s.n.c. by Th34; :: thesis: verum

end;

end;