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 :: thesis: ( Index (p,f) = Index (q,f) & not LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) implies LE q,p,f /. (Index (q,f)),f /. ((Index (q,f)) + 1) )

A7: Index (p,f) < len f by A2, Th8;
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_3:25;
A10: (Index (p,f)) + 0 <> (Index (p,f)) + 1 ;
A11: 1 <= Index (p,f) by A2, Th8;
then A12: LSeg (f,(Index (p,f))) = LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A8, TOPREAL1:def 3;
then A13: p in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A2, Th9;
Index (p,f) in dom f by A11, A7, FINSEQ_3:25;
then A14: f /. (Index (p,f)) <> f /. ((Index (p,f)) + 1) by A1, A9, A10, PARTFUN2:10;
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, Th9;
then LT q,p,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) by A16, A13, A14, Th28;
hence LE q,p,f /. (Index (q,f)),f /. ((Index (q,f)) + 1) by A15; :: 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, Def7;
then A18: Rev (B_Cut (f,q,p)) = B_Cut (f,p,q) by A2, A3, A17, A6, Def7;
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, Th15; :: thesis: verum

end;

end;