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 vertical &
  r in LSeg(p,q) holds p`1 = r`1
proof
  let p,q,r be Point of TOP-REAL 2;
  assume LSeg(p,q) is vertical;
  then
A1: p`1 = q`1 by Th16;
  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`1 = (1-t)*p`1+t*p`1 .= ((1-t)*p)`1+t*q`1 by A1,TOPREAL3:4
    .= ((1-t)*p)`1+(t*q)`1 by TOPREAL3:4
    .= r`1 by A2,TOPREAL3:2;
end;
