reserve n,i,k,m for Nat;
reserve r,r1,r2,s,s1,s2 for Real;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL 2,
  f,f1,f2 for FinSequence of the carrier of TOP-REAL 2,
  p,p1,p2,p3,q,q3 for Point of TOP-REAL 2;

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;
