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 Th17:
  (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 f, b be Element of E such that
        A1: x = f + b & f in (v+A) & b in B;
        consider a be Element of E such that
        A2: f = v + a & a in A by A1;
        A3: x = a+ (v + b) by A1,A2,RLVECT_1:def 3;
        v+b in (v+B) by A1;
        hence x in A(+)(v + B) by A3,A2;
      end;

      assume x in A(+)(v + B);
      then consider a, f be Element of E such that
      A4: x = a + f & a in A & f in (v + B);
      consider b be Element of E such that
      A5: f=v+b & b in B by A4;
      A6: x = (v+a) + b by A4,A5,RLVECT_1:def 3;
      v+a in (v+A) by A4;
      hence x in (v+A)(+)B by A6,A5;
    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 f, b be Element of E such that
        A7: x = f+b & f in (v+A) & b in B;
        consider a be Element of E such that
        A8: f = v+a & a in A by A7;
        A9: x = v + (a+b) by A7,A8,RLVECT_1:def 3;
        a+b in (A+B) by A7,A8;
        hence x in v+ (A(+)B) by 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);
      consider a, b be Element of E such that
      A11: ab = a + b & a in A & b in B by A10;
      A12: x = (v+a) + b by A10,A11,RLVECT_1:def 3;
      v+a in (v+A) by A11;
      hence x in (v + A)(+)B by A12,A11;
    end;
    hence (v+A)(+)B = v + (A(+)B) by TARSKI:2;
  end;
