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+q`1)/2,p`2]| in LSeg(p,q)
proof
  set p1 = |[(p`1+q`1)/2,p`2]|;
  assume that
A1: p`1 <> q`1 and
A2: p`2 = q`2;
A3: p = |[p`1,p`2]| & q = |[q`1,p`2]| by A2,EUCLID:53;
A4: p1`1 = (p`1+q`1)/2 & p1`2 = p`2;
  per cases by A1,XXREAL_0:1;
  suppose
A5: p`1 < q`1;
    then p`1 <= (p`1+q`1)/2 & (p`1+q`1)/2 <= q`1 by XREAL_1:226;
    then p1 in {p2: p2`2 = p`2 & p`1 <= p2`1 & p2`1 <= q`1} by A4;
    hence thesis by A3,A5,Th10;
  end;
  suppose
A6: p`1 > q`1;
    then q`1 <= (p`1+q`1)/2 & (p`1+q`1)/2 <= p`1 by XREAL_1:226;
    then p1 in {p2: p2`2 = p`2 & q`1 <= p2`1 & p2`1 <= p`1} by A4;
    hence thesis by A3,A6,Th10;
  end;
end;
