
theorem Th27:
  for f being FinSequence of TOP-REAL 2 st f is being_S-Seq
  holds L_Cut (f,f/.1) = f
proof
  let f be FinSequence of TOP-REAL 2;
  assume
A1: f is being_S-Seq;
  then
A2: 1+1 <= len f by TOPREAL1:def 8;
  then 1 <= len f by XXREAL_0:2;
  then
A3: 1 in dom f by FINSEQ_3:25;
A4: 1+1 in dom f by A2,FINSEQ_3:25;
A5: 1 < len f by A2,NAT_1:13;
A6: f is one-to-one by A1;
A7: f/.1 = f.1 by A3,PARTFUN1:def 6;
A8: Index(f/.1,f) = 1 by A2,JORDAN3:11;
  f/.1 <> f/.(1+1) by A3,A4,A6,PARTFUN2:10;
  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 A8,JORDAN3:def 3
    .= mid(f,1,len f) by A3,A5,A7,A8,FINSEQ_6:126
    .= f by A2,FINSEQ_6:120,XXREAL_0:2;
end;
