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)).((the carrier of E) \ A)
  = (the carrier of E) \ ((erosion(C)).A)
  & (erosion(C)).( (the carrier of E) \ A)
  = (the carrier of E) \ ((dilation(C)).A )
  proof
    thus (dilation(C)).((the carrier of E) \ A)
    = ((the carrier of E) \ A)(+)C by Def2
    .= (the carrier of E) \ (A(-)C) by Th9
    .= (the carrier of E) \ ((erosion(C)).A) by Def3;
    thus (erosion(C)).( (the carrier of E) \ A)
    = ((the carrier of E) \ A)(-)C by Def3
    .= (the carrier of E) \ (A(+)C) by Th9
    .= (the carrier of E) \ ((dilation(C)).A) by Def2;
  end;
