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`1 = q`1 & p`2 <> q`2 implies |[p`1,(p`2+q`2)/2]| in LSeg(p,q)
proof
  set p1 = |[p`1,(p`2+q`2)/2]|;
  assume that
A1: p`1 = q`1 and
A2: p`2 <> q`2;
A3: p = |[p`1,p`2]| & q = |[p`1,q`2]| by A1,EUCLID:53;
A4: p1`1 = p`1 & p1`2 = (p`2+q`2)/2;
  per cases by A2,XXREAL_0:1;
  suppose
A5: p`2 < q`2;
    then p`2 <= (p`2+q`2)/2 & (p`2+q`2)/2 <= q`2 by XREAL_1:226;
    then p1 in {p2: p2`1 = p`1 & p`2 <= p2`2 & p2`2 <= q`2} by A4;
    hence thesis by A3,A5,Th9;
  end;
  suppose
A6: p`2 > q`2;
    then q`2 <= (p`2+q`2)/2 & (p`2+q`2)/2 <= p`2 by XREAL_1:226;
    then p1 in {p2: p2`1 = p`1 & q`2 <= p2`2 & p2`2 <= p`2} by A4;
    hence thesis by A3,A6,Th9;
  end;
end;
