reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem
  for p1,p2,q1,q2 being Point of TOP-REAL 2 st LE q1,q2,p1,p2 & LE q2,q1
  ,p1,p2 holds q1=q2
proof
  let p1,p2,q1,q2 be Point of TOP-REAL 2;
  assume that
A1: LE q1,q2,p1,p2 and
A2: LE q2,q1,p1,p2;
  q1 in LSeg(p1,p2) by A1;
  then consider r1 such that
A3: q1 = (1-r1)*p1+r1*p2 and
A4: 0<=r1 and
A5: r1<=1;
  q2 in LSeg(p1,p2) by A1;
  then consider r2 such that
A6: q2 = (1-r2)*p1+r2*p2 and
A7: 0<=r2 and
A8: r2<=1;
A9: r2<=r1 by A2,A3,A4,A5,A6,A8;
  r1<=r2 by A1,A3,A5,A6,A7,A8;
  then r1=r2 by A9,XXREAL_0:1;
  hence thesis by A3,A6;
end;
