
theorem Th42:
  for a,b,c being Real
  for u,v,w being Element of TOP-REAL 3
  st a <> 0 & a + b + c = 0 &
  a * u + b * v + c * w = 0.TOP-REAL 3 holds
  u in Line(v,w)
  proof
    let a,b,c be Real;
    let u,v,w be Element of TOP-REAL 3;
    assume that
A1: a <> 0 and
A2: a + b + c = 0 and
A3: a * u + b * v + c * w = 0.TOP-REAL 3;
A4: u = ((-b)/a) * v + ((-c)/a) * w by A1,A3,ANPROJ_8:12;
    reconsider r = (-c) / a as Real;
    (1 - r) = a / a + c / a by A1,XCMPLX_1:60
           .= (a + c) / a
           .= (-b) / a by A2;
    then u in the set of all (1 - r) * v + r * w where r is Real by A4;
    hence thesis by RLTOPSP1:def 14;
  end;
