let f be non empty FinSequence of (TOP-REAL 2); :: thesis: for p being Point of (TOP-REAL 2) st f is weakly-one-to-one & len f >= 2 holds
L_Cut (f,(f /. 1)) = f
let p be Point of (TOP-REAL 2); :: thesis: ( f is weakly-one-to-one & len f >= 2 implies L_Cut (f,(f /. 1)) = f )
assume that
A1: f is weakly-one-to-one and
A2: len f >= 2 ; :: thesis: L_Cut (f,(f /. 1)) = f
A3: Index ((f /. 1),f) = 1 by A2, JORDAN3:11;
1 <= len f by A2, XXREAL_0:2;
then A4: 1 in dom f by FINSEQ_3:25;
then A5: f /. 1 = f . 1 by PARTFUN1:def 6;
2 = 1 + 1 ;
then A6: 1 < len f by A2, NAT_1:13;
then f . 1 <> f . (1 + 1) by A1;
then f /. 1 <> f . (1 + 1) by A4, PARTFUN1:def 6;
hence L_Cut (f,(f /. 1)) = <*(f /. 1)*> ^ (mid (f,((Index ((f /. 1),f)) + 1),(len f))) by A3, JORDAN3:def 3
.= mid (f,1,(len f)) by A4, A6, A5, A3, FINSEQ_6:126
.= f by A2, FINSEQ_6:120, XXREAL_0:2 ;
:: thesis: verum