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;
