
theorem
  for V being RealUnitarySpace, v,w being VECTOR of V st v in Lin{w} & v
  <> 0.V holds Lin{v} = Lin{w}
proof
  let V be RealUnitarySpace;
  let v,w be VECTOR of V;
  assume that
A1: v in Lin{w} and
A2: v <> 0.V;
  consider a be Real such that
A3: v = a * w by A1,Th29;
  now
    let u be VECTOR of V;
A4: a <> 0 by A2,A3,RLVECT_1:10;
    thus u in Lin{v} implies u in Lin{w}
    proof
      assume u in Lin{v};
      then consider b being Real such that
A5:   u = b * v by Th29;
      u = b * a * w by A3,A5,RLVECT_1:def 7;
      hence thesis by Th29;
    end;
    assume u in Lin{w};
    then consider b being Real such that
A6: u = b * w by Th29;
    a" * v = a" * a * w by A3,RLVECT_1:def 7
      .= 1 * w by A4,XCMPLX_0:def 7
      .= w by RLVECT_1:def 8;
    then u = b * a" * v by A6,RLVECT_1:def 7;
    hence u in Lin{v} by Th29;
  end;
  hence thesis by RUSUB_1:25;
end;
