reserve r,s,t,u for Real;
reserve V for RealLinearSpace,
  v,w for Point of V;

theorem :: JORDAN2C:47
  0.V in LSeg(v,w) implies ex r st v = r*w or w = r*v
proof
  assume 0.V in LSeg(v,w);
  then consider s being Real such that
A1: 0.V= (1-s)*v + s*w and
  0 <= s and
  s <= 1;
  -s*w=(1-s)*v by A1,RLVECT_1:6;
  then
A2: (-s)*w =(1-s)*v by RLVECT_1:79;
  per cases;
  suppose
A3: -s<>0;
    take r = (-s)"*(1-s);
    w = 1*w by RLVECT_1:def 8
      .= (-s)"*(-s)*w by A3,XCMPLX_0:def 7
      .= (-s)"*((-s)*w) by RLVECT_1:def 7
      .= r*v by A2,RLVECT_1:def 7;
    hence thesis;
  end;
  suppose
A4: -s=0;
    take -s;
    thus thesis by A2,A4,RLVECT_1:def 8;
  end;
end;
