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
  p1 in LSeg(p,q) & p2 in LSeg(p,q) & p1`1<>p2`1 & p1`2=p2`2 implies
  LSeg(p,q) is horizontal
proof
  assume p1 in LSeg(p,q);
  then consider r1 such that
A1: p1 = (1-r1)*p+r1*q and
  0<=r1 and
  r1<=1;
  assume p2 in LSeg(p,q);
  then consider r2 such that
A2: p2 = (1-r2)*p+r2*q and
  0<=r2 and
  r2<=1;
  assume that
A3: p1`1 <> p2`1 and
A4: p1`2 = p2`2;
  p`2-(r1*(p`2)-r1*(q`2))= (1-r1)*(p`2)+r1*(q`2)
    .= (1-r1)*(p`2)+(r1*q)`2 by TOPREAL3:4
    .= ((1-r1)*p)`2+(r1*q)`2 by TOPREAL3:4
    .= p1`2 by A1,TOPREAL3:2
    .= ((1-r2)*p)`2+(r2*q)`2 by A2,A4,TOPREAL3:2
    .= (1-r2)*(p`2)+(r2*q)`2 by TOPREAL3:4
    .= 1*(p`2)-r2*(p`2)+r2*(q`2) by TOPREAL3:4
    .= p`2-(r2*(p`2)-r2*(q`2));
  then
A5: (r1-r2)*(p`2)=(r1-r2)*(q`2);
  r1-r2<>0 by A1,A2,A3;
  then p`2 = q`2 by A5,XCMPLX_1:5;
  hence thesis by Th15;
end;
