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 Th25:
  (A(-)B)(-)C = A(-)(B(+)C)
  proof
    for x be object holds x in (A(-)B)(-)C iff x in A(-)(B(+)C)
    proof
      let x be object;
      hereby
        assume x in (A(-)B)(-)C;
        then consider w be Element of E such that
        A1: x = w & for c be Element of E st c in C
        holds w - c in (A(-)B);
        now
          let bc be Element of E;
          assume bc in (B(+)C);
          then consider b, c be Element of E such that
          A2: bc = b+c & b in B & c in C;
          w - c in (A(-)B) by A1,A2;
          then consider g be Element of E such that
          A3: w - c = g & for b be Element of E st b in B holds g - b in A;
          w - bc = g - b by A2,A3,RLVECT_1:27;
          hence w - bc in A by A3,A2;
        end;
        hence x in A(-)(B(+)C) by A1;
      end;
      assume x in A(-)(B(+)C);
      then consider w be Element of E such that
      A4: x = w & for bc be Element of E st bc in B(+)C holds w - bc in A;
      now let c be Element of E;
        assume
        A5: c in C;
        now let b be Element of E;
          assume
          A6: b in B;
          b+c in B(+)C by A5,A6;
          then w -(b+c) in A by A4;
          hence (w -c) -b in A by RLVECT_1:27;
        end;
        hence w-c in A(-)B;
      end;
      hence x in (A(-)B)(-)C by A4;
    end;
    hence (A(-)B)(-)C = A(-)(B(+)C) by TARSKI:2;
  end;
