reserve E for RealLinearSpace;
reserve A, B, C for binary-image of E;
reserve a, b, v for Element of E;
reserve F, G for binary-image-family of E;
reserve A, B, C for non empty binary-image of E;

theorem
  A(+)(union F) = union {A(+)X where X is binary-image of E: X in F}
  proof
    A1: for x be object
    holds x in {X(+)A where X is binary-image of E: X in F}
    iff x in {A(+)X where X is binary-image of E: X in F}
    proof
      let x be object;
      hereby assume x in {X(+)A where X is binary-image of E: X in F};
        then consider W be binary-image of E such that
        A2: x = W(+)A & W in F;
        x = A(+)W & W in F by A2;
        hence x in {A(+)X where X is binary-image of E: X in F};
      end;

      assume x in {A(+)X where X is binary-image of E: X in F};
      then consider W be binary-image of E such that
      A3: x= A(+)W & W in F;
      x= W(+)A & W in F by A3;
      hence x in {X(+)A where X is binary-image of E: X in F};
    end;
    thus A(+)(union F) = (union F)(+)A
    .= union {X(+)A where X is binary-image of E: X in F} by Th12
    .= union {A(+)X where X is binary-image of E: X in F}
    by A1,TARSKI:2;
  end;
