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 Th15:
  1 <= i & i+1 <= len f & p in LSeg(f/.i,f/.(i+1)) implies p in L~ f
proof
  assume that
A1: 1 <= i and
A2: i+1 <= len f and
A3: p in LSeg(f/.i,f/.(i+1));
  set X = { LSeg(f,j) : 1 <= j & j+1 <= len f };
A4: LSeg(f,i) in X by A1,A2;
  LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A1,A2,TOPREAL1:def 3;
  hence thesis by A3,A4,TARSKI:def 4;
end;
