BYPASSING NATURALS FOR INTEGERS
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   CLOSURE ON NATURAL NUMBER SUBTRACTION
          --------------------->
     SHIFT|                    ^CLOSURE ON
      FROM|                    |DEFINED
   NATURAL|                    |DIFFERENCE
   NUMBERS|                    |SUBTRACTION
          |                    |(RESTRICTED
          V-------------------->SYSTEM)
 DEVELOP DEFINED DIFFERENCE ARITHMETIC
 MINUEND @ m; SUBTRAHEND @ s;
DFIND DIFFERENCE:  [m - s]  s. t.  m < s;
DIFF. CLOSURE: for OPERATION  o  and DEF. DIFFERENCE  d :   [d1]  o  [d] = [d3] (COMBINED DEF. DIFFS IS DEF. DIFF):
 [m1 - s1] > [m2 - s2] <--> m1 + s2 >m2 + s1;
 [m1 - s1] < [m2 - s2] <--> m1 + s2 < m2 + s1;
 [m1 - s1] = [m2 - s2] <--> m1 + s2 = m2 + s1;
 [m1 - s1] + [m2 - s2] = [m1 + m2] - [s1 + s2];
 [m1 - s1] - [m2 - s2] = [m1 + s2] - [m2 + s2];
 [m1 - s1] * [m2 - s2] = [m1 * m2 + s1 * s2 ] -
 [m1 * s2 + m2 * s1]
       CLOSURE ON A TOTAL NUMBER SYSTEM       
          ------------------------->
SHIFT FROM|                        ^ADDITION &
   DEFINED|                        |SUBTRACTION
DIFFERENCE|                        |CLOSED IN
ARITHMETIC|                        |TOTAL NUMBER
          |                        |SYSTEM
          V----------------------- >
     DEVELOP NATURAL VECTORS ARITHMETIC
         COMBINING
   1st VECTOR COMP. = u   2nd VEC. COMP. = v
VECTOR: [u, v] s.t. the COMPS are NATURALS;
ARITH. OF VECTORS OF NATURAL NUMBERS:
 [u1, v1] > [u2, v2] <--> u1 + v2 >u2 + v1;
 [u1, v1] < [u2, v2] <--> u1 + v2 < u2 + v1;
 [u1, v1] = [u2, v2] <--> u1 + v2 = u2 + v1;
 [u1, v1] + [u2, v2] = [u1 + u2], [v1 + v2];
 [u1, v1] - [u2, v2] = [u1 + v2,u2 + v1];
 [u1,v1] * [u2,v2] = [u1*u2 + v1*v2, u1*v2 + u2*v1];
WHEREAS DEFINED DIFFENCES HAVE RESTRICTION ON COMPONENTS, NO SUCH RESTRICTION ON VECTOR COMPONENTS, PROVIDED NO VIOLATING OF NATURAL NO. ARITHMETIC, HOWEVER, NOTE THAT SUBTRACTION ON VECTORS BECOMES ADDITION ON COMPONENTS, HENCE ALWAYS ALLOWED! THUS, CLOSURE GOAL ACHIEVED FOR VECTOR SUBTRACTION! 
        "AWKWARD" VECTOR NOTATION       
        ------------------------->USE POSITIVE,
   SHIFT|                        ^NEGATIVE SIGNS,
    FROM|                        |ZERO: 3 CLASSES
 VECTORS|                        |OF INTEGERS
      OF|                        |CLOSED FOR
NATURALS|                        |ADDITION &
        V----------------------- >SUBTRACTION
  REDUCE: 3 EQUIVALENCE CLASSES OF VECTORS
         COMBINING
 SO "INTEGERS" ARE VECTORS OF NATURAL NUMBERS, BUT THE VECTOR NOTATION IS AWKWARD & CAN BE BYPASSED. HOW? BY THE VECTOR EQUIVALENCE RULE:  [u1, v1] = [u2, v2] <--> u1 + v2 = u2 + v1;
LET u > v IN A VECTOR, SO THAT u - v = w; THEN THIS NATURAL NUMBER SUBTRACTION IS ALLOWED: [u, v] = [u - v, v - v] = [w, 0]: CALL THIS A "POSITIVE VECTOR"; ALTERNATIVELY, LET u < v, SO THAT v - u = x, THEN [u - u, v - u] = [0, x]: CALL THIS A "NEGATIVE VECTOR"; OR, IF u = v, [u - u, v - v] = [0,0]: CALL THIS THE NULL VECTOR, SIMPLY DENOTED AS 0 INTEGER. THEN THE POSITVE VECTOR CAN BE HIDDEN BY A SIGN: +w; THEN NEGATIVE VECTOR BY ANOTHER SIGN: -x. AND WE HAVE OUR FAMILIAR INTEGER NOTATION (HIDING VECTORS OF NATURALS!), PROVIDING CLOSURE FOR SUBTRACTION -- WITHOUT CHEATING!
HOWEVER, SOMETHING SPECIAL OCCURRED ABOVE. WE CAN BEST INVOKE IT BY APPLYING THE VECTOR PRODUCT RULE ( [u1,v1] * [u2,v2] = [u1*u2 + v1*v2, u1*v2 + u2*v1]) TO POSITIVE AND NEGATIVE UNIT VECTORS: [1,0] * [1,0] = [1 * 1 + 0 * 0, 1 * 0 + 0 * 1] = [1,0]: POSITIVE; AND [0,1] * [0,1] = [0 * 0 + 1 * 1, 0 * 1 + 1 * 0] = [1,0]: ALSO POSITIVE; WHEREAS [1,0] * [0,1] = [1 * 0 + 0 * 1, 0 * 0 + 1 * 1] = [0,1]: NEGATIVE! HEY! THIS IS THAT "WEIRD" RULE OF SIGNS FOR VECTORS. BUT WHENCE CAME IT? BACK THERE IN NATURAL NUMBER ARITHMETIC WHEN WE REQUIRED CLOSURE ON OPERATING WITH DEFINED DIFFERENCES!

IN ORDER TO OBTAIN A MULTIPICATION RULE FOR DEFINED DIFFERENCES THAT YIELDED A DEFINED DIFFERENCE ANSWER, WE HAD TO ADOPT THE FORM SUCH THAT DEFINED DIFFERENCE TIMES DEFINED DIFFERENCE EQUALS A DEFINED DIFFERENCE:  [m1 - s1] * [m2 - s2] = [m1 * m2 + s1 * s2] -  [m1 * s2 + m2 * s1]. AND WHEN WE USE THIS TO MODEL VECTORS OF NATURALS, THOSE VECTORS PRODUCTS (POS. TIMES POS. IS POS.; NEG. TIMES NEG. IS NEG; POS. TIMES NEG. = NEG., AND VISY VERSY!) POP OUT!

NO ONE DECIDED TO MAKE UP A WEIRD RULE OF SIGNS. IT FOLLOWED FROM THE MOST "SACRED" RULE IN ARITHMETIC -- PERHAPS IN ALL MATH! -- CLOSURE. DIG?