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
  (dilation(B)).(union F)
  = union {(dilation(B)).X where X is binary-image of E: X in F}
  proof
A1: for x be object holds x in {X(+)B where X is binary-image of E: X in F}
    iff x in {(dilation(B)).X where X is binary-image of E: X in F}
    proof
      let x be object;

      hereby assume x in {X(+)B where X is binary-image of E: X in F};
        then consider X be binary-image of E such that
        A2: x = X(+)B & X in F;
        x = (dilation(B)).X & X in F by A2,Def2;
        hence x in {(dilation(B)).W where W is binary-image of E: W in F};
      end;

      assume x in {(dilation(B)).X where X is binary-image of E: X in F};
      then consider X be binary-image of E such that
      A3: x=(dilation(B)).X & X in F;
      x = X(+)B & X in F by A3,Def2;
      hence x in { W(+)B where W is binary-image of E: W in F};
    end;
    thus (dilation(B)).(union F) = (union F)(+)B by Def2
    .= union { X(+)B where X is binary-image of E: X in F} by Th12
    .= union {(dilation(B)).X where X is binary-image of E: X in F}
    by A1,TARSKI:2;
  end;
