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 Th15:
  p`2 = q`2 iff LSeg(p,q) is horizontal
proof
  set P = LSeg(p,q);
  thus p`2=q`2 implies P is horizontal
  proof
    assume
A1: p`2=q`2;
    let p1,p2;
    assume
A2: p1 in P;
    assume p2 in P;
    then
A3: p`2 <= p2`2 & p2`2 <= p`2 by A1,TOPREAL1:4;
    p`2 <= p1`2 & p1`2 <= p`2 by A1,A2,TOPREAL1:4;
    then p`2 = p1`2 by XXREAL_0:1;
    hence thesis by A3,XXREAL_0:1;
  end;
  p in P & q in P by RLTOPSP1:68;
  hence thesis;
end;
