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;
reserve P,Q for Subset of TOP-REAL 2,
  f,f1,f2 for FinSequence of the carrier of TOP-REAL 2,
  p,p1,p2,p3,q,q3 for Point of TOP-REAL 2;

theorem
  for p,q,r being Point of TOP-REAL 2 st LSeg(p,q) is horizontal & r in
  LSeg(p,q) holds p`2 = r`2
proof
  let p,q,r be Point of TOP-REAL 2;
  assume LSeg(p,q) is horizontal;
  then
A1: p`2 = q`2 by Th15;
  assume r in LSeg(p,q);
  then consider t being Real such that
A2: r = (1-t)*p+t*q and
  0 <= t and
  t <= 1;
  thus p`2 = (1-t)*p`2+t*p`2 .= ((1-t)*p)`2+t*q`2 by A1,TOPREAL3:4
    .= ((1-t)*p)`2+(t*q)`2 by TOPREAL3:4
    .= r`2 by A2,TOPREAL3:2;
end;
