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
  union {X(-)B where X is binary-image of E: X in F} c= (union F)(-)B
  proof
    let x be object;
    assume x in union { X(-)B where X is binary-image of E: X in F};
    then consider Y be set such that
    A1: x in Y & Y in {X(-)B where X is binary-image of E: X in F}
    by TARSKI:def 4;
    consider W be binary-image of E such that
    A2: Y = W(-)B & W in F by A1;
    consider z be Element of E such that
    A3: x = z & for b be Element of E st b in B holds z - b in W by A1,A2;
    now let b be Element of E;
      assume b in B;
      then
      A4: z - b in W by A3;
      W c= union F by A2,ZFMISC_1:74;
      hence z - b in union F by A4;
    end;
    hence x in (union F) (-)B by A3;
  end;
