let f be FinSequence of (TOP-REAL 2); :: thesis: for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . (len f) holds
L_Cut f,p is_S-Seq_joining p,f /. (len f)
let p be Point of (TOP-REAL 2); :: thesis: ( f is being_S-Seq & p in L~ f & p <> f . (len f) implies L_Cut f,p is_S-Seq_joining p,f /. (len f) )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: p <> f . (len f) ; :: thesis: L_Cut f,p is_S-Seq_joining p,f /. (len f)
A4: f <> {} by A2, CARD_1:47, TOPREAL1:28;
A5: Rev f is being_S-Seq by A1, SPPOL_2:47;
A6: p in L~ (Rev f) by A2, SPPOL_2:22;
A7: p <> (Rev f) . 1 by A3, FINSEQ_5:65;
L_Cut f,p = L_Cut (Rev (Rev f)),p by FINSEQ_6:29
.= Rev (R_Cut (Rev f),p) by A1, A6, Th57, SPPOL_2:47 ;
then L_Cut f,p is_S-Seq_joining p,(Rev f) /. 1 by A5, A6, A7, Th48, Th67;
hence L_Cut f,p is_S-Seq_joining p,f /. (len f) by A4, FINSEQ_5:68; :: thesis: verum