theorem Th16:
  p`1 = q`1 iff LSeg(p,q) is vertical
proof
  set P = LSeg(p,q);
  thus p`1=q`1 implies P is vertical
  proof
    assume
A1: p`1=q`1;
    let p1,p2;
    assume
A2: p1 in P;
    assume p2 in P;
    then
A3: p`1 <= p2`1 & p2`1 <= p`1 by A1,TOPREAL1:3;
    p`1 <= p1`1 & p1`1 <= p`1 by A1,A2,TOPREAL1:3;
    then p`1 = p1`1 by XXREAL_0:1;
    hence thesis by A3,XXREAL_0:1;
  end;
  p in P & q in P by RLTOPSP1:68;
  hence thesis;
end;
