let f be FinSequence of (TOP-REAL 2); :: thesis: ( f is special & f is weakly-one-to-one implies for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f holds
B_Cut f,p,q is special )
assume A1: ( f is special & f is weakly-one-to-one ) ; :: thesis: for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f holds
B_Cut f,p,q is special
let p, q be Point of (TOP-REAL 2); :: thesis: ( p in L~ f & q in L~ f implies B_Cut f,p,q is special )
assume that
A2: p in L~ f and
A3: q in L~ f ; :: thesis: B_Cut f,p,q is special
A4: Index p,f < len f by A2, JORDAN3:41;
A5: 1 <= Index p,f by A2, JORDAN3:41;
then A6: Index p,f in dom f by A4, FINSEQ_3:27;

per cases ( p <> q or p = q ) ;

suppose A7: p <> q ; :: thesis: B_Cut f,p,q is special

now

per cases ( ( p in L~ f & q in L~ f & 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 ( p in L~ f & q in L~ f & 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 A8: ( p in L~ f & q in L~ f & Index p,f < Index q,f ) ; :: thesis: B_Cut f,p,q is special

then A9: q in L~ (L_Cut f,p) by JORDAN3:64;
L_Cut f,p is special by A1, A2, Th29;
then R_Cut (L_Cut f,p),q is special by A9, Th30;
hence B_Cut f,p,q is special by A8, JORDAN3:def 8; :: thesis: verum

end;

suppose A10: ( 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 special

A11: L_Cut f,p is special by A1, A2, Th29;
q in L~ (L_Cut f,p) by A2, A3, A7, A10, JORDAN3:66;
then R_Cut (L_Cut f,p),q is special by A11, Th30;
hence B_Cut f,p,q is special by A10, JORDAN3:def 8; :: thesis: verum

end;

suppose A12: ( not ( p in L~ f & q in L~ f & 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 special

A13: now

per cases ( Index q,f < Index p,f or Index q,f = Index p,f ) by A2, A3, A12, XXREAL_0:1;

suppose Index q,f < Index p,f ; :: thesis: p in L~ (L_Cut f,q)

hence p in L~ (L_Cut f,q) by A2, A3, JORDAN3:64; :: thesis: verum

end;

suppose A14: Index q,f = Index p,f ; :: thesis: p in L~ (L_Cut f,q)

A15: (Index p,f) + 1 >= 1 by NAT_1:11;
(Index p,f) + 1 <= len f by A4, NAT_1:13;
then A16: (Index p,f) + 1 in dom f by A15, FINSEQ_3:27;
set Ls = LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1));
A17: q in LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A3, A14, JORDAN5B:32;
A18: p in LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) by A2, JORDAN5B:32;
f . (Index p,f) <> f . ((Index p,f) + 1) by A1, A5, A4, Def2;
then f . (Index p,f) <> f /. ((Index p,f) + 1) by A16, PARTFUN1:def 8;
then A19: f /. (Index p,f) <> f /. ((Index p,f) + 1) by A6, PARTFUN1:def 8;
then A20: LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)) is_an_arc_of f /. (Index p,f),f /. ((Index p,f) + 1) by TOPREAL1:15;
not LE p,q, LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)),f /. (Index p,f),f /. ((Index p,f) + 1) by A12, A14, A19, JORDAN5C:17;
then LE q,p, LSeg (f /. (Index p,f)),(f /. ((Index p,f) + 1)),f /. (Index p,f),f /. ((Index p,f) + 1) by A7, A20, A18, A17, JORDAN5C:14;
hence p in L~ (L_Cut f,q) by A2, A3, A7, A14, A19, JORDAN3:66, JORDAN5C:17; :: thesis: verum

end;

end;

end;

A21: B_Cut f,p,q = Rev (R_Cut (L_Cut f,q),p) by A12, JORDAN3:def 8;
L_Cut f,q is special by A1, A3, Th29;
then R_Cut (L_Cut f,q),p is special by A13, Th30;
hence B_Cut f,p,q is special by A21, SPPOL_2:42; :: thesis: verum

end;

end;

end;

hence B_Cut f,p,q is special ; :: thesis: verum

end;

suppose p = q ; :: thesis: B_Cut f,p,q is special

then B_Cut f,p,q = <*p*> by A1, A2, Th15;
hence B_Cut f,p,q is special by SPPOL_2:39; :: thesis: verum

end;

end;