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 Th32:
  p`1 <> q`1 & p`2 = q`2 implies LSeg(p,|[(p`1+q`1)/2,p`2]|) /\
  LSeg(|[(p`1+q`1)/2,p`2]|,q)={|[(p`1+q`1)/2,p`2]|}
proof
  assume that
A1: p`1 <> q`1 and
A2: p`2 = q`2;
  set p3 = |[(p`1+q`1)/2,p`2]|;
  set l23 = LSeg(p,p3), l = LSeg(p3,q);
  thus l23 /\ l c= {p3}
  proof
    let x be object;
A3: l23 = LSeg(|[p`1,p`2]|,|[(p`1+q`1)/2,p`2]|) by EUCLID:53;
    assume
A4: x in l23 /\ l;
    then
A5: x in l by XBOOLE_0:def 4;
A6: l = LSeg(|[(p`1+q`1)/2,q`2]|,|[q`1,q`2]|) by A2,EUCLID:53;
A7: x in l23 by A4,XBOOLE_0:def 4;
    now
      per cases by A1,XXREAL_0:1;
      suppose
A8:     p`1 < q`1;
        then p`1 < (p`1+q`1)/2 by XREAL_1:226;
        then x in {p1: p1 `2 = p`2 & p`1 <= p1 `1 & p1 `1 <= (p`1+q`1)/2} by A7
,A3,Th10;
        then consider t1 be Point of TOP-REAL 2 such that
A9:     t1 = x and
A10:    t1 `2 = p`2 and
        p`1 <= t1 `1 and
A11:    t1 `1 <= (p`1+q`1)/2;
A12:    t1`1 <= p3 `1 by A11;
        (p`1+q`1)/2 < q`1 by A8,XREAL_1:226;
        then x in {p2: p2 `2 = q`2 & (p`1+q`1)/2 <= p2 `1 & p2 `1 <= q`1} by A5
,A6,Th10;
        then
        ex t2 be Point of TOP-REAL 2 st t2 = x & t2 `2 = q`2 & (p`1+q`1)/
        2 <= t2 `1 & t2 `1 <= q`1;
        then t1`1 >= p3 `1 by A9;
        then
A13:    t1 `1 = p3 `1 by A12,XXREAL_0:1;
        t1`2 = p3 `2 by A10;
        hence x=p3 by A9,A13,Th6;
      end;
      suppose
A14:    p`1 > q`1;
        then p`1 > (p`1+q`1)/2 by XREAL_1:226;
        then x in {p11: p11 `2 = p`2 & (p`1+q`1)/2 <= p11 `1 & p11 `1 <= p`1}
        by A7,A3,Th10;
        then consider t1 be Point of TOP-REAL 2 such that
A15:    t1 = x and
A16:    t1 `2 = p`2 and
A17:    (p`1+q`1)/2 <= t1 `1 and
        t1 `1 <= p`1;
A18:    p3 `1 <= t1 `1 by A17;
        q`1 < (p`1+q`1)/2 by A14,XREAL_1:226;
        then x in {p22: p22 `2 = q`2 & q`1<=p22 `1 & p22 `1<= (p`1+q`1)/2} by
A5,A6,Th10;
        then ex t2 be Point of TOP-REAL 2 st t2 = x & t2 `2 = q`2 & q`1 <= t2
        `1 & t2 `1 <= (p`1+q`1)/2;
        then t1`1 <= p3 `1 by A15;
        then
A19:    t1 `1 = p3 `1 by A18,XXREAL_0:1;
        t1`2 = p3 `2 by A16;
        hence x=p3 by A15,A19,Th6;
      end;
    end;
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {p3};
  then
A20: x=p3 by TARSKI:def 1;
  p3 in l23 & p3 in l by RLTOPSP1:68;
  hence thesis by A20,XBOOLE_0:def 4;
end;
