reserve P,P1,P2,R for Subset of TOP-REAL 2,
  p,p1,p2,p3,p11,p22,q,q1,q2,q3,q4 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r for Real,
  u for Point of Euclid 2,
  n,m,i,j,k for Nat,
  x,y for set;

theorem Th18:
  p<>f/.1 & f is being_S-Seq & p in L~f implies ex h st h is
being_S-Seq & h/.1=f/.1 & h/.len h = p & L~h is_S-P_arc_joining f/.1,p & L~h c=
  L~f
proof
  set M = {LSeg(f,i): 1<=i & i+1<=len f}, p1 = f/.1;
  assume that
A1: p<>p1 & f is being_S-Seq and
A2: p in L~f;
  consider X be set such that
A3: p in X and
A4: X in M by A2,TARSKI:def 4;
  consider n such that
A5: X=LSeg(f,n) and
  1<=n and
  n+1<=len f by A4;
  consider h such that
A6: h is being_S-Seq & h/.1=p1 & h/.len h = p & L~h is_S-P_arc_joining
  p1,p & L~h c= L~f and
  L~h = L~(f|n) \/ LSeg(f/.n,p) by A1,A3,A5,Th17;
  take h;
  thus thesis by A6;
end;
