reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;

theorem Th6:
  for p1,p2,q1,q2 being Point of TOP-REAL n st q1 in LSeg(p1,p2) &
  q2 in LSeg(p1,p2) holds LSeg(q1,q2) c= LSeg(p1,p2)
proof
  let p1,p2,q1,q2 be Point of TOP-REAL n;
  assume that
A1: q1 in LSeg(p1,p2) and
A2: q2 in LSeg(p1,p2);
A3: LSeg(p1,p2) = LSeg(p1,q1) \/ LSeg(q1,p2) by A1,Th5;
  now
    per cases by A2,A3,XBOOLE_0:def 3;
    suppose
A4:   q2 in LSeg(p1,q1);
A5:   LSeg(p1,q1) c= LSeg(p1,p2) by A1,Lm1;
      LSeg(q2,q1) c= LSeg(p1,q1) by A4,Lm1;
      hence thesis by A5;
    end;
    suppose
A6:   q2 in LSeg(q1,p2);
A7:   LSeg(q1,p2) c= LSeg(p1,p2) by A1,Lm1;
      LSeg(q1,q2) c= LSeg(q1,p2) by A6,Lm1;
      hence thesis by A7;
    end;
  end;
  hence thesis;
end;
