let f be FinSequence of (TOP-REAL 2); :: thesis: for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & f is being_S-Seq holds
L~ (B_Cut f,p,q) c= L~ f
let p, q be Point of (TOP-REAL 2); :: thesis: ( p in L~ f & q in L~ f & f is being_S-Seq implies L~ (B_Cut f,p,q) c= L~ f )
assume that
A1: p in L~ f and
A2: q in L~ f and
A3: f is being_S-Seq ; :: thesis: L~ (B_Cut f,p,q) c= L~ f
A4: f is one-to-one by A3, TOPREAL1:def 10;

per cases ( p = q or ( p <> q & ( Index p,f < Index q,f or ( Index p,f = Index q,f & LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) ) or ( p <> q & not Index p,f < Index q,f & not ( Index p,f = Index q,f & LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) ) ;

suppose p = q ; :: thesis: L~ (B_Cut f,p,q) c= L~ f

then B_Cut f,p,q = <*p*> by A1, A4, Th24;
then len (B_Cut f,p,q) = 1 by FINSEQ_1:56;
then L~ (B_Cut f,p,q) = {} by TOPREAL1:28;
hence L~ (B_Cut f,p,q) c= L~ f by XBOOLE_1:2; :: thesis: verum

end;

suppose ( p <> q & ( Index p,f < Index q,f or ( Index p,f = Index q,f & LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) ) ; :: thesis: L~ (B_Cut f,p,q) c= L~ f

hence L~ (B_Cut f,p,q) c= L~ f by A1, A2, Lm4; :: thesis: verum

end;

suppose that A5: p <> q and
A6: ( not Index p,f < Index q,f & not ( Index p,f = Index q,f & LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) ; :: thesis: L~ (B_Cut f,p,q) c= L~ f

A7: B_Cut f,p,q = Rev (R_Cut (L_Cut f,q),p) by A6, JORDAN3:def 8;
A8: ( Index q,f < Index p,f or ( Index p,f = Index q,f & not LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) by A6, XXREAL_0:1;

A9: now

assume that
A10: Index p,f = Index q,f and
A11: not LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ; :: thesis: LE q,p,f /. (Index q,f),f /. ((Index q,f) + 1)
A12: 1 <= Index p,f by A1, JORDAN3:41;
A13: Index p,f < len f by A1, JORDAN3:41;
then A14: (Index p,f) + 1 <= len f by NAT_1:13;
then A15: LSeg f,(Index p,f) = LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A12, TOPREAL1:def 5;
then A16: p in LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A1, JORDAN3:42;
A17: q in LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A2, A10, A15, JORDAN3:42;
A18: Index p,f in dom f by A12, A13, FINSEQ_3:27;
1 <= (Index p,f) + 1 by NAT_1:11;
then A19: (Index p,f) + 1 in dom f by A14, FINSEQ_3:27;
(Index p,f) + 0 <> (Index p,f) + 1 ;
then f /. (Index p,f) <> f /. ((Index p,f) + 1) by A4, A18, A19, PARTFUN2:17;
then LT q,p,f /. (Index p,f),f /. ((Index p,f) + 1) by A11, A16, A17, JORDAN3:63;
hence LE q,p,f /. (Index q,f),f /. ((Index q,f) + 1) by A10, JORDAN3:def 7; :: thesis: verum

end;

then A20: Rev (B_Cut f,q,p) = B_Cut f,p,q by A1, A2, A7, A8, JORDAN3:def 8;
L~ (B_Cut f,q,p) c= L~ f by A1, A2, A5, A8, A9, Lm4;
hence L~ (B_Cut f,p,q) c= L~ f by A20, SPPOL_2:22; :: thesis: verum

end;

end;