reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;
reserve n for Nat,
  p,q,q1,q2 for Point of TOP-REAL 2,
  r,s1,s2,t1,t2 for Real,
  x,y for Point of Euclid 2;

theorem
  q1 in LSeg(q2,p) & q1 <> q2 implies dist(q1,p) < dist(q2,p)
proof
  assume that
A1: q1 in LSeg(q2,p) and
A2: q1 <> q2;
  dist(q2,q1) + dist(q1,p) = dist(q2,p) by A1,Th29;
  hence thesis by A2,Th22,XREAL_1:29;
end;
