let f be FinSequence of (TOP-REAL 2); :: thesis: ( 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) /. 1 = p )
assume A1: 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) /. 1 = p
let p, q be Point of (TOP-REAL 2); :: thesis: ( p in L~ f & q in L~ f implies (B_Cut f,p,q) /. 1 = p )
assume that
A2: p in L~ f and
A3: q in L~ f ; :: thesis: (B_Cut f,p,q) /. 1 = p
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;
A7: 1 <= len (L_Cut f,p) by A2, Th1;

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

suppose A8: p <> q ; :: thesis: (B_Cut f,p,q) /. 1 = p

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 or not q in L~ f or not Index p,f < Index q,f ) & ( not Index p,f = Index q,f or not LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) ) ;

suppose A9: ( p in L~ f & q in L~ f & Index p,f < Index q,f ) ; :: thesis: (B_Cut f,p,q) /. 1 = p

then q in L~ (L_Cut f,p) by JORDAN3:64;
then (R_Cut (L_Cut f,p),q) /. 1 = (L_Cut f,p) /. 1 by SPRECT_3:39
.= (L_Cut f,p) . 1 by A7, FINSEQ_4:24
.= p by A9, JORDAN3:58 ;
hence (B_Cut f,p,q) /. 1 = p by A9, 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) /. 1 = p

then q in L~ (L_Cut f,p) by A2, A3, A8, JORDAN3:66;
then (R_Cut (L_Cut f,p),q) /. 1 = (L_Cut f,p) /. 1 by SPRECT_3:39
.= (L_Cut f,p) . 1 by A7, FINSEQ_4:24
.= p by A2, JORDAN3:58 ;
hence (B_Cut f,p,q) /. 1 = p by A10, JORDAN3:def 8; :: thesis: verum

end;

suppose A11: ( ( not p in L~ f or not q in L~ f or not Index p,f < Index q,f ) & ( not Index p,f = Index q,f or not LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) ; :: thesis: (B_Cut f,p,q) /. 1 = p

then A12: B_Cut f,p,q = Rev (R_Cut (L_Cut f,q),p) by JORDAN3:def 8;

now

per cases ( Index p,f > Index q,f or Index p,f = Index q,f ) by A2, A3, A11, XXREAL_0:1;

suppose Index p,f > Index q,f ; :: thesis: (B_Cut f,p,q) /. 1 = p

then A13: p in L~ (L_Cut f,q) by A2, A3, JORDAN3:64;
R_Cut (L_Cut f,q),p <> {} by JORDAN1J:44;
hence (B_Cut f,p,q) /. 1 = (R_Cut (L_Cut f,q),p) /. (len (R_Cut (L_Cut f,q),p)) by A12, FINSEQ_5:68
.= p by A13, JORDAN1J:45 ;
:: thesis: verum

end;

suppose A14: Index p,f = Index q,f ; :: thesis: (B_Cut f,p,q) /. 1 = p

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 A11, 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 A8, A20, A18, A17, JORDAN5C:14;
then A21: p in L~ (L_Cut f,q) by A2, A3, A8, A14, A19, JORDAN3:66, JORDAN5C:17;
R_Cut (L_Cut f,q),p <> {} by JORDAN1J:44;
hence (B_Cut f,p,q) /. 1 = (R_Cut (L_Cut f,q),p) /. (len (R_Cut (L_Cut f,q),p)) by A12, FINSEQ_5:68
.= p by A21, JORDAN1J:45 ;
:: thesis: verum

end;

end;

end;

hence (B_Cut f,p,q) /. 1 = p ; :: thesis: verum

end;

end;

end;

hence (B_Cut f,p,q) /. 1 = p ; :: thesis: verum

end;

suppose p = q ; :: thesis: (B_Cut f,p,q) /. 1 = p

then B_Cut f,p,q = <*p*> by A1, A2, Th15;
hence (B_Cut f,p,q) /. 1 = p by FINSEQ_4:25; :: thesis: verum

end;

end;