reserve n for Nat;

theorem Th15: :: JORDAN5B:24
  for f be FinSequence of TOP-REAL 2 for p be Point of TOP-REAL 2
  st p in L~f & f is weakly-one-to-one holds B_Cut(f,p,p) = <*p*>
proof
  let f be FinSequence of TOP-REAL 2;
  let p be Point of TOP-REAL 2;
  assume that
A1: p in L~f and
A2: f is weakly-one-to-one;
A3: 1 <= Index(p,f) by A1,JORDAN3:8;
  L_Cut(f,p).1 = p by A1,JORDAN3:23;
  then
A4: R_Cut(L_Cut(f,p),p) = <*p*> by JORDAN3:def 4;
A5: Index(p,f) < len f by A1,JORDAN3:8;
  then
A6: Index(p,f)+1 <= len f by NAT_1:13;
  then
A7: Index(p,f) + 1 in dom f by A3,SEQ_4:134;
  f.Index(p,f) <> f.(Index(p,f)+1) by A2,A5,A3;
  then
A8: f.Index(p,f) <> f/.(Index(p,f)+1) by A7,PARTFUN1:def 6;
  Index(p,f) in dom f by A3,A6,SEQ_4:134;
  then
A9: f/.Index(p,f) <> f/.(Index(p,f)+1) by A8,PARTFUN1:def 6;
  p in LSeg(f,Index(p,f)) by A1,JORDAN3:9;
  then p in LSeg(f/.Index(p,f), f/.(Index(p,f)+1)) by A3,A6,TOPREAL1:def 3;
  then LE p,p,f/.Index(p,f),f/.(Index(p,f)+1) by A9,JORDAN5B:9;
  hence thesis by A4,JORDAN3:def 7;
end;
