reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;

theorem Th27:
  for P being Subset of TOP-REAL n, w1,w2,w3,w4,w5,w6,w7 being
Point of TOP-REAL n st w1 in P & w2 in P & w3 in P & w4 in P & w5 in P & w6 in
P & w7 in P & LSeg(w1,w2) c= P & LSeg(w2,w3) c= P & LSeg(w3,w4) c= P & LSeg(w4,
  w5) c= P & LSeg(w5,w6) c= P & LSeg(w6,w7) c= P ex h being Function of I[01],(
  TOP-REAL n) |P st h is continuous & w1=h.0 & w7=h.1
proof
  let P be Subset of TOP-REAL n, w1,w2,w3,w4,w5,w6,w7 be Point of TOP-REAL n;
  assume that
A1: w1 in P and
A2: w2 in P & w3 in P & w4 in P & w5 in P & w6 in P & w7 in P & LSeg(w1,
w2) c= P & LSeg(w2,w3) c= P & LSeg(w3,w4) c= P & LSeg(w4,w5) c= P & LSeg(w5, w6
  ) c= P & LSeg(w6,w7) c= P;
  ( ex h2 being Function of I[01],(TOP-REAL n) |P st h2 is continuous & w1=
  h2.0 & w4=h2.1)& ex h4 being Function of I[01],(TOP-REAL n) |P st h4 is
  continuous & w4=h4.0 & w7=h4.1 by A1,A2,Th26;
  hence thesis by A1,Th24;
end;
