reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th14:
  p in L~f implies ex i st 1 <= i & i+1 <= len f & p in LSeg(f/.i, f/.(i+1))
proof
  assume p in L~f;
  then consider i such that
A1: 1 <= i and
A2: i+1 <= len f and
A3: p in LSeg(f,i) by Th13;
  take i;
  thus 1 <= i & i+1 <= len f by A1,A2;
  thus thesis by A1,A2,A3,TOPREAL1:def 3;
end;
