
theorem Th22:
  for f being FinSequence of TOP-REAL 2, p, q being Point of TOP-REAL 2 st
  p in L~f & q in L~f & q <> f.len f & p = f.len f & f is being_S-Seq holds
  p in L~L_Cut(f,q)
proof
  let f be FinSequence of TOP-REAL 2, p, q be Point of TOP-REAL 2;
  assume that
A1: p in L~f and
A2: q in L~f and
A3: q <> f.len f and
A4: p = f.len f and
A5: f is being_S-Seq;
  1 + 1 <= len f by A5,TOPREAL1:def 8;
  then
A6: 1 < len f by XXREAL_0:2;
  then
A7: Index(p,f) + 1 = len f by A4,A5,JORDAN3:12;
AA: len f in dom f by A6,FINSEQ_3:25; then
AB: mid(f,len f,len f) = <*f.len f*> by FINSEQ_6:193
       .= <*f/.len f*> by AA,PARTFUN1:def 6;
  Index(q,f) < len f by A2,JORDAN3:8;
  then
A8: Index(q,f) <= Index(p,f) by A7,NAT_1:13;
  per cases by A8,XXREAL_0:1;
  suppose Index (q,f) = Index (p,f);
    then
A9: L_Cut(f,q) = <*q*>^mid(f,len f,len f) by A3,A7,JORDAN3:def 3
      .= <*q*>^<*f/.len f*> by AB
      .= <*q,f/.len f*> by FINSEQ_1:def 9
      .= <*q,p*> by A4,A6,FINSEQ_4:15;
    then rng L_Cut(f,q) = {p,q} by FINSEQ_2:127;
    then
A10: p in rng L_Cut(f,q) by TARSKI:def 2;
    len L_Cut(f,q) = 2 by A9,FINSEQ_1:44;
    then rng L_Cut(f,q) c= L~L_Cut(f,q) by SPPOL_2:18;
    hence thesis by A10;
  end;
  suppose Index (q,f) < Index (p,f);
    hence thesis by A1,A2,JORDAN3:29;
  end;
end;
