reserve V for non empty RealLinearSpace;

theorem Lm02:
  for X be VectSp of F_Real holds
    RLSStruct (# the carrier of X, the ZeroF of X,
                 the addF of X, MultReal*(X) #) is RealLinearSpace
proof
  let X be VectSp of F_Real;
  set XQ=RLSStruct (# the carrier of X, the ZeroF of X,
                      the addF of X, MultReal*(X) #);
A1: for vZ1,wZ1 be Element of X, v,w be Element of XQ
      st v = vZ1 & w = wZ1 holds v + w = vZ1 + wZ1;
  XQ is Abelian add-associative right_zeroed
    right_complementable scalar-distributive vector-distributive
    scalar-associative scalar-unital
  proof
   hereby let v,w be VECTOR of XQ;
    reconsider vZ1 = v, wZ1 = w as Element of X;
    thus v + w = wZ1 + vZ1 by A1
                .= w + v;
   end;
   hereby let u,v,w be VECTOR of XQ;
    reconsider uZ1 = u, vZ1 = v, wZ1 = w as Element of X;
    (u + v) + w = (uZ1 + vZ1) + wZ1
               .= uZ1 + (vZ1 + wZ1) by RLVECT_1:def 3;
    hence (u + v) + w = u + (v + w);
   end;
   hereby let v be VECTOR of XQ;
    reconsider vZ1 = v as Element of X;
    thus v + 0.XQ = vZ1 + 0.X .= v;
   end;
   thus XQ is right_complementable
   proof
    let v be VECTOR of XQ;
    reconsider vZ1 = v as Element of X;
    consider wZ1 be Element of X such that
A2:  vZ1 + wZ1 = 0.X by ALGSTR_0:def 11;
    reconsider w = wZ1 as VECTOR of XQ;
    take w;
    thus v + w = 0.XQ by A2;
   end;
   hereby let a,b be Real, v be VECTOR of XQ;
    reconsider vZ1=v as Element of X;
    reconsider aZ1=a, bZ1=b as Element of F_Real by XREAL_0:def 1;
    (a + b) * v = (aZ1 + bZ1) * vZ1
               .= aZ1* vZ1 + bZ1* vZ1 by VECTSP_1:def 15;
    hence (a + b) * v = a * v + b * v;
   end;
   hereby let a be Real, v,w be VECTOR of XQ;
    reconsider aZ1=a as Element of F_Real by XREAL_0:def 1;
    reconsider vZ1 = v, wZ1 = w as Element of X;
    a * (v + w) = aZ1*(vZ1+wZ1)
               .= aZ1*vZ1 + aZ1*wZ1 by VECTSP_1:def 14;
    hence a * (v + w) =a*v + a*w;
   end;
   hereby let a,b be Real, v be VECTOR of XQ;
    reconsider vZ1=v as Element of X;
    reconsider aZ1=a, bZ1=b as Element of F_Real by XREAL_0:def 1;
    (a * b) * v = (aZ1 * bZ1) * vZ1
               .= aZ1 * (bZ1 * vZ1) by VECTSP_1:def 16;
    hence (a * b) * v = a * (b * v);
   end;
   let v be VECTOR of XQ;
   reconsider vZ1=v as Element of X;
   thus 1 * v = 1.(F_Real) *vZ1
              .= v;
  end;
  hence thesis;
end;
