reserve p,p1,p2,p3,p11,p22,q,q1,q2 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r,r1,r2,s,s1,s2 for Real,
  u,u1,u2,u5 for Point of Euclid 2,
  n,m,i,j,k for Nat,
  N for Nat,
  x,y,z for set;
reserve lambda for Real;

theorem
  p in LSeg(|[r1,r]|,|[s1,r]|) implies p`2 = r
proof
  assume
A1: p in LSeg(|[r1,r]|,|[s1,r]|);
  per cases by XXREAL_0:1;
  suppose
    r1<s1;
    then LSeg(|[r1,r]|,|[s1,r]|) = {q: q`2 = r & r1<=q`1 & q`1<=s1} by Th10;
    then ex p1 st p1=p & p1`2 = r & r1<=p1`1 & p1`1<=s1 by A1;
    hence thesis;
  end;
  suppose
    r1=s1;
    then p in {|[r1,r]|} by A1,RLTOPSP1:70;
    then p = |[r1,r]| by TARSKI:def 1;
    hence thesis;
  end;
  suppose
    r1>s1;
    then LSeg(|[r1,r]|,|[s1,r]|) = {q: q`2 = r & s1<=q`1 & q`1<=r1} by Th10;
    then ex p1 st p1=p & p1`2 = r & s1<=p1`1 & p1`1<=r1 by A1;
    hence thesis;
  end;
end;
