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 Th24:
  (v+A)(-)B = A(-)(v+B) & (v+A)(-)B = v+(A(-)B)
  proof
    for x be object holds x in (v+A)(-)B iff x in A(-)(v+B)
    proof
      let x be object;
      hereby
        assume x in (v+A)(-) B;
        then consider w be Element of E such that
        A1: x = w & for b be Element of E st b in B holds w - b in (v+A);
        now let vb be Element of E;
          assume vb in (v+B); then
          consider b be Element of E such that
          A2: vb = v+b & b in B;
          w - b in (v+A) by A2,A1;
          then consider a be Element of E such that
          A3: w - b = v+a & a in A;
          w - vb = w-b -v by A2,RLVECT_1:27
          .= a by A3,RLVECT_4:1;
          hence w - vb in A by A3;
        end;
        hence x in A(-)(v+B) by A1;
      end;
      assume x in A(-)(v+B);
      then consider w be Element of E such that
      A4: x = w & for vb be Element of E st vb in (v+B) holds w - vb in A;
      now let b be Element of E;
        assume b in B;
        then (v+b) in (v+B);
        then w -(v+b) in A by A4;
        then
        A5: v + (w -(v+b)) in v+A;
        v + (w -(v+b)) = v+w -(v+b) by RLVECT_1:28
        .=w+(v-(v+b)) by RLVECT_1:28
        .=w+ (v -v - b) by RLVECT_1:27
        .=w + (0.E - b) by RLVECT_1:15;
        hence w - b in v+A by A5;
      end;
      hence x in (v+A)(-) B by A4;
    end;
    hence (v+A)(-) B = A(-) (v+B) by TARSKI:2;

    for x be object holds x in (v+A)(-) B iff x in v+ (A(-)B)
    proof
      let x be object;
      hereby
        assume x in (v+A)(-) B;
        then consider w be Element of E such that
        A6: x = w & for b be Element of E st b in B holds w - b in (v+A);
        now let b be Element of E;
          assume b in B;
          then
          A7: w - b in (v+A) by A6;
          consider a be Element of E such that
          A8: w - b  = v+a & a in A by A7;
          (w - v) - b = w - (v+b) by RLVECT_1:27
          .= v+a -v by A8,RLVECT_1:27
          .= a by RLVECT_4:1;
          hence (w - v) - b in A by A8;
        end;
        then
        A9: w-v in A(-)B;
        v+(w-v) = w by RLVECT_4:1;
        hence x in v + (A(-)B) by A6,A9;
      end;

      assume x in v + (A(-)B);
      then consider ab be Element of E such that
      A10: x = v+ab & ab in (A(-)B);

      reconsider w=x as Element of E by A10;
      consider d be Element of E such that
      A11: ab = d & for b be Element of E st b in B holds d - b in A by A10;

      now let b be Element of E;
        assume b in B;
        then
        A12: ab - b in A by A11;
        (v+ab) - b = v +(ab-b) by RLVECT_1:28;
        hence (v+ab) - b in v + A by A12;
      end;
      hence x in (v+A)(-)B by A10;
    end;
    hence (v+A)(-)B = v+(A(-)B) by TARSKI:2;
  end;
