let f be FinSequence of (TOP-REAL 2); :: thesis: for p, q being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & q in L~ f & p <> q holds
B_Cut f,p,q is_S-Seq_joining p,q
let p, q be Point of (TOP-REAL 2); :: thesis: ( f is being_S-Seq & p in L~ f & q in L~ f & p <> q implies B_Cut f,p,q is_S-Seq_joining p,q )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: q in L~ f and
A4: p <> q ; :: thesis: B_Cut f,p,q is_S-Seq_joining p,q

per cases ( 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 ( 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 ( 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: B_Cut f,p,q is_S-Seq_joining p,q

hence B_Cut f,p,q is_S-Seq_joining p,q by A1, A2, A3, A4, Lm1; :: thesis: verum

end;

suppose A5: ( 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: B_Cut f,p,q is_S-Seq_joining p,q

A6: now

A7: Index p,f < len f by A2, Th41;
then A8: (Index p,f) + 1 <= len f by NAT_1:13;
1 <= (Index p,f) + 1 by NAT_1:11;
then A9: (Index p,f) + 1 in dom f by A8, FINSEQ_3:27;
A10: (Index p,f) + 0 <> (Index p,f) + 1 ;
A11: 1 <= Index p,f by A2, Th41;
then A12: LSeg f,(Index p,f) = LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A8, TOPREAL1:def 5;
then A13: p in LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A2, Th42;
Index p,f in dom f by A11, A7, FINSEQ_3:27;
then A14: f /. (Index p,f) <> f /. ((Index p,f) + 1) by A1, A9, A10, PARTFUN2:17;
assume that
A15: Index p,f = Index q,f and
A16: 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)
q in LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A3, A15, A12, Th42;
then LT q,p,f /. (Index p,f),f /. ((Index p,f) + 1) by A16, A13, A14, Th63;
hence LE q,p,f /. (Index q,f),f /. ((Index q,f) + 1) by A15, Def7; :: thesis: verum

end;

A17: ( 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 A5, XXREAL_0:1;
B_Cut f,p,q = Rev (R_Cut (L_Cut f,q),p) by A5, Def8;
then A18: Rev (B_Cut f,q,p) = B_Cut f,p,q by A2, A3, A17, A6, Def8;
B_Cut f,q,p is_S-Seq_joining q,p by A1, A2, A3, A4, A17, A6, Lm1;
hence B_Cut f,p,q is_S-Seq_joining p,q by A18, Th48; :: thesis: verum

end;

end;