reserve V for non empty RealLinearSpace;

theorem Lm01:
  for X be RealLinearSpace holds
    ModuleStr (# the carrier of X, the addF of X, the ZeroF of X,
                 MultF_Real*(X) #)
      is VectSp of F_Real
proof
  let X be RealLinearSpace;
  set XP=ModuleStr (# the carrier of X, the addF of X, the ZeroF of X,
                      MultF_Real*(X) #);
Q1:XP is scalar-distributive vector-distributive
         scalar-associative scalar-unital
  proof
   now let x,y be Element of F_Real;
    let v be Element of XP;
    reconsider v1=v as Element of X;
    reconsider x1=x,y1=y as Real;
    (x+y)*v = (x1+y1)*v1
            = x1*v1+y1*v1 by RLVECT_1:def 6;
    hence (x+y)*v = x*v+y*v;
   end;
   hence XP is scalar-distributive;
   now let x be Element of F_Real;
    let v,w be Element of XP;
    reconsider v1=v,w1=w as Element of X;
    reconsider x1=x as Real;
    x*(v+w) = x1*(v1+w1)
           .= x1*v1+x1*w1 by RLVECT_1:def 5;
    hence x*(v+w) = x*v+x*w;
   end;
   hence XP is vector-distributive;
   now let x,y be Element of F_Real;
    let v be Element of XP;
    reconsider v1=v as Element of X;
    reconsider x1=x,y1=y as Real;
    (x*y)*v = (x1*y1)*v1
           .= x1*(y1*v1) by RLVECT_1:def 7;
    hence (x*y)*v = x*(y*v);
   end;
   hence XP is scalar-associative;
   now let v be Element of XP;
    reconsider v1=v as Element of X;
    (1.F_Real)*v = 1*v1;
    hence (1.F_Real)*v = v by RLVECT_1:def 8;
   end;
   hence XP is scalar-unital;
  end;
  now let u,v,w be Element of XP;
   reconsider u1=u,v1=v,w1=w as Element of X;
   u+(v + w) = u1+(v1+w1)
            .= (u1+v1)+w1 by RLVECT_1:def 3;
   hence u+(v + w) = (u+v)+w;
  end; then
Q2: XP is add-associative;
  now let v be Element of XP;
   reconsider v1=v as Element of X;
   v + 0.XP = v1+0.X;
   hence v + 0.XP = v;
  end; then
Q3: XP is right_zeroed;
  now let v be Element of XP;
   reconsider v1=v as Element of X;
   consider w1 be Element of X such that
A1: v1+w1= 0.X by ALGSTR_0:def 11;
   reconsider w=w1 as Element of XP;
   v + w = 0.XP by A1;
   hence v is right_complementable;
  end;
  then
Q4: XP is right_complementable;
  now let v,w be Element of XP;
   reconsider v1=v,w1=w as Element of X;
   v + w = v1+w1 .= w1+v1;
   hence v + w = w+v;
  end;
  then XP is Abelian;
  hence XP is VectSp of F_Real by Q1,Q2,Q3,Q4;
end;
