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
  (erosion(B)).(meet F)
  = meet {(erosion(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 {(erosion(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 = (erosion(B)).X & X in F by A2,Def3;
        hence x in {(erosion(B)).W where W is binary-image of E: W in F};
      end;

      assume x in {(erosion(B)).X where X is binary-image of E: X in F};
      then consider X be binary-image of E such that
      A3: x = (erosion(B)).X & X in F;

      x = X(-)B & X in F by A3,Def3;
      hence x in { W(-)B where W is binary-image of E: W in F};
    end;
    thus (erosion(B)).(meet F) = (meet F)(-)B by Def3
    .= meet {X(-)B where X is binary-image of E: X in F } by Th18
    .= meet {(erosion(B)).X where X is binary-image of E: X in F}
    by A1,TARSKI:2;
  end;
