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(C)).((dilation(B)).A) = (dilation((dilation(C)).B)).A
  & (erosion(C)).((erosion(B)).A) = (erosion((dilation(C)).B)).A
  proof
    thus (dilation(C)).((dilation(B)).A) = (dilation(C)).(A(+)B) by Def2
    .= (A(+)B)(+)C by Def2
    .= A(+)(B(+)C) by Th10
    .= A(+)((dilation(C)).B) by Def2
    .= (dilation((dilation(C)).B)).A by Def2;
    thus (erosion(C)).((erosion(B)).A) = (erosion(C)).(A(-)B) by Def3
    .= (A(-)B)(-)C by Def3
    .= A(-)(B(+)C) by Th25
    .= A(-)((dilation(C)).B) by Def2
    .= (erosion((dilation(C)).B)).A by Def3;
  end;
