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
  R_Algebra_of_BoundedFunctions X is RealLinearSpace
proof
  now
    let v being VECTOR of R_Algebra_of_BoundedFunctions X;
    reconsider v1=v as VECTOR of RAlgebra X by TARSKI:def 3;
    R_Algebra_of_BoundedFunctions X is Subalgebra of RAlgebra X by Th6;
    then jj * v = jj*v1 by Th8;
    hence 1 * v =v by FUNCSDOM:12;
  end;
  hence thesis by Lm2;
end;
