reserve n for Nat;

theorem
  for f be non empty FinSequence of TOP-REAL 2 for p be Point of
  TOP-REAL 2 st f is weakly-one-to-one & len f >= 2 holds L_Cut (f,f/.1) = f
proof
  let f be non empty FinSequence of TOP-REAL 2;
  let p be Point of TOP-REAL 2;
  assume that
A1: f is weakly-one-to-one and
A2: len f >= 2;
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;
end;
