let f be FinSequence of (TOP-REAL 2); :: thesis: for p being Point of (TOP-REAL 2) st p in L~ f & f is one-to-one holds
B_Cut (f,p,p) = <*p*>
let p be Point of (TOP-REAL 2); :: thesis: ( p in L~ f & f is one-to-one implies B_Cut (f,p,p) = <*p*> )
assume that
A1: p in L~ f and
A2: f is one-to-one ; :: thesis: B_Cut (f,p,p) = <*p*>
A3: Index (p,f) <> (Index (p,f)) + 1 ;
A4: Index (p,f) < len f by A1, JORDAN3:8;
A5: 1 <= Index (p,f) by A1, JORDAN3:8;
A6: (Index (p,f)) + 1 <= len f by A4, NAT_1:13;
then A7: Index (p,f) in dom f by A5, SEQ_4:134;
A8: (Index (p,f)) + 1 in dom f by A5, A6, SEQ_4:134;
p in LSeg (f,(Index (p,f))) by A1, JORDAN3:9;
then p in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A5, A6, TOPREAL1:def 3;
then A9: LE p,p,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) by A2, A3, A7, A8, Th9, PARTFUN2:10;
(L_Cut (f,p)) . 1 = p by A1, JORDAN3:23;
then R_Cut ((L_Cut (f,p)),p) = <*p*> by JORDAN3:def 4;
hence B_Cut (f,p,p) = <*p*> by A9, JORDAN3:def 7; :: thesis: verum