
theorem Th11:
  for p, p1, p2 being Point of TOP-REAL 2 st p in LSeg(p1,p2) & p1 <> p2
  holds LE p,p2,p1,p2
proof
  let p, p1, p2 be Point of TOP-REAL 2;
  assume that
A1: p in LSeg(p1,p2) and
A2: p1 <> p2;
  thus LE p,p2,p1,p2
  proof
    thus p in LSeg(p1,p2) & p2 in LSeg(p1,p2) by A1,RLTOPSP1:68;
    let r1,r2 be Real such that
    0<=r1 and
A3: r1<=1 and p=(1-r1)*p1+r1*p2
    and 0<=r2
    and r2<=1 and
A4: p2=(1-r2)*p1+r2*p2;
    p2 = 1*p2 by RLVECT_1:def 8
      .= 0.TOP-REAL 2 + 1*p2 by RLVECT_1:4
      .= (1-1)*p1 + 1*p2 by RLVECT_1:10;
    hence thesis by A2,A3,A4,JORDAN5A:2;
  end;
end;
