
theorem Th7:
  for l being RLSStruct st the RLSStruct of l is RealLinearSpace
  holds l is RealLinearSpace
proof
  let l be RLSStruct such that
A1: the RLSStruct of l is RealLinearSpace;
  reconsider l as non empty RLSStruct by A1;
  reconsider l0=RLSStruct (# the carrier of l, 0.l, the addF of l, the Mult of
    l #) as RealLinearSpace by A1;
A2: l is Abelian
  proof
    let v,w be VECTOR of l;
    reconsider v1=v as VECTOR of l0;
    reconsider w1=w as VECTOR of l0;
    thus v+w =v1+w1 .= w1+v1
      .= w +v;
  end;
A3: l is right_zeroed
  proof
    let v be VECTOR of l;
    reconsider v1=v as VECTOR of l0;
    thus v+0.l= v1 + 0.l0 .= v;
  end;
A4: l is right_complementable
  proof
    let v be VECTOR of l;
    reconsider v1=v as VECTOR of l0;
    consider w1 being VECTOR of l0 such that
A5: v1 + w1 = 0.l0 by ALGSTR_0:def 11;
    reconsider w = w1 as VECTOR of l;
    take w;
    thus thesis by A5;
  end;
A6: for v being VECTOR of l holds 1 * v = v
  proof
    let v be VECTOR of l;
    reconsider v1=v as VECTOR of l0;
    thus 1*v= 1*v1 .= v by RLVECT_1:def 8;
  end;
A7: for a,b be Real for v being VECTOR of l holds (a * b) * v = a *
  (b * v)
  proof
    let a,b be Real;
    let v be VECTOR of l;
    reconsider v1=v as VECTOR of l0;
    thus (a*b)*v =(a*b)*v1 .=a*(b*v1) by RLVECT_1:def 7
      .= a*(b*v);
  end;
A8: for a,b be Real for v being VECTOR of l holds (a + b) * v = a *
  v + b * v
  proof
    let a,b be Real;
    let v be VECTOR of l;
    reconsider v1=v as VECTOR of l0;
    thus (a+b)*v =(a+b)*v1 .=a*v1+b*v1 by RLVECT_1:def 6
      .= a*v +b*v;
  end;
A9: for a be Real for v,w being VECTOR of l holds a * (v + w) = a *
  v + a * w
  proof
    let a be Real;
    let v,w be VECTOR of l;
    reconsider v1=v, w1=w as VECTOR of l0;
    thus a*(v+w) =a*(v1+w1) .=a*v1+a*w1 by RLVECT_1:def 5
      .= a*v +a*w;
  end;
  l is add-associative
  proof
    let u,v,w be VECTOR of l;
    reconsider u1=u, v1=v, w1=w as VECTOR of l0;
    thus (u + v) + w = (u1+v1)+w1 .= u1+(v1+w1) by RLVECT_1:def 3
      .= u+(v+w);
  end;
  hence thesis by A2,A3,A4,A9,A8,A7,A6,RLVECT_1:def 5,def 6,def 7,def 8;
end;
