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 Th2:
  for p1,p2,q1,q2 being Point of TOP-REAL 2 st (p1`1 = p2`1 or p1
  `2 = p2`2)& q1 in LSeg(p1,p2) & q2 in LSeg(p1,p2) holds q1`1 = q2`1 or q1`2 =
  q2`2
proof
  let p1,p2,q1,q2 be Point of TOP-REAL 2;
  assume that
A1: p1`1 = p2`1 or p1`2 = p2`2 and
A2: q1 in LSeg(p1,p2) and
A3: q2 in LSeg(p1,p2);
  consider r2 such that
A4: q2= (1-r2)*p1 + r2*p2 and
  0 <= r2 and
  r2 <= 1 by A3;
  consider r1 such that
A5: q1= (1-r1)*p1 + r1*p2 and
  0 <= r1 and
  r1 <= 1 by A2;
  q1`1=((1-r1)*p1)`1+(r1*p2)`1 by A5,TOPREAL3:2;
  then q1`1=(1-r1)*(p1`1)+(r1*p2)`1 by TOPREAL3:4;
  then
A6: q1`1=(1-r1)*(p1`1)+r1*(p2`1) by TOPREAL3:4;
  q2`1=((1-r2)*p1)`1+(r2*p2)`1 by A4,TOPREAL3:2;
  then q2`1=(1-r2)*(p1`1)+(r2*p2)`1 by TOPREAL3:4;
  then
A7: q2`1=(1-r2)*(p1`1)+r2*(p2`1) by TOPREAL3:4;
  q1`2=((1-r1)*p1)`2+(r1*p2)`2 by A5,TOPREAL3:2;
  then q1`2=(1-r1)*(p1`2)+(r1*p2)`2 by TOPREAL3:4;
  then
A8: q1`2=(1-r1)*(p1`2)+r1*(p2`2) by TOPREAL3:4;
  q2`2=((1-r2)*p1)`2+(r2*p2)`2 by A4,TOPREAL3:2;
  then q2`2=(1-r2)*(p1`2)+(r2*p2)`2 by TOPREAL3:4;
  then
A9: q2`2=(1-r2)*(p1`2)+r2*(p2`2) by TOPREAL3:4;
  per cases by A1;
  suppose
    p1`1 = p2`1;
    hence thesis by A6,A7;
  end;
  suppose
    p1`2 = p2`2;
    hence thesis by A8,A9;
  end;
end;
