reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;

theorem
  for V being Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative
  vector-associative non empty AlgebraStr st V is scalar-mult-cancelable
  holds V is RealLinearSpace
proof
  let V being Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative
  vector-associative non empty AlgebraStr;
  assume
A1: V is scalar-mult-cancelable;
 for v being VECTOR of V holds 1 * v = v
  proof
    let v be VECTOR of V;
    (1*v) + (1 * -v) = 1 * (v + -v) by RLVECT_1:def 5;
    then (1*v) - (1*v) = 0.V & (1*v) + (1 * -v) = 1 * 0.V by RLVECT_1:5;
    then
A2: -(jj*v) = jj*(-v) by Th4,RLVECT_1:8;
    1 * v = (1 * 1) * v .= 1 * (1 * v) by RLVECT_1:def 7;
    then 1 * (1 * v) - 1 * v = 0.V by RLVECT_1:15;
    then 1 * (1 * v - v) = 0.V by A2,RLVECT_1:def 5;
    then (jj*v) - v = 0.V by A1;
    hence thesis by RLVECT_1:21;
  end;
  hence thesis by RLVECT_1:def 8;
end;
