reserve n,i,k,m for Nat;
reserve r,r1,r2,s,s1,s2 for Real;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL n;

theorem
  for p1,p2,q1,q2 st LSeg(p1,p2) = LSeg(q1,q2) holds
  p1 = q1 & p2 = q2 or p1 = q2 & p2 = q1
proof
  let p1,p2,q1,q2;
A1: q1 in LSeg(q1,q2) by RLTOPSP1:68;
A2: q2 in LSeg(q1,q2) by RLTOPSP1:68;
  assume
A3: LSeg(p1,p2)=LSeg(q1,q2);
  per cases by A3,Th7,RLTOPSP1:68;
  suppose
A4: p1=q1 & p2=q1;
    then LSeg(p1,p2) = {q1} by RLTOPSP1:70;
    hence thesis by A3,A2,A4,TARSKI:def 1;
  end;
  suppose
A5: p1=q2 & p2=q2;
    then LSeg(p1,p2) = {q2} by RLTOPSP1:70;
    hence thesis by A3,A1,A5,TARSKI:def 1;
  end;
  suppose
    p1=q1 & p2=q2 or p1=q2 & p2=q1;
    hence thesis;
  end;
end;
