FOR THE STUDENT OF MATHEMATICS
DEMONSTRATION CLUB FILE XXXXII
PURPOSE: To demonstrate that for real number x, real number -x, real number y, real number -y and real number z (in the note):

                    XXXXII.     - ( x + y ) = -x - y

is true.

DEMONSTRATION:      STATEMENTS                                           REASONS



      1.     ( x + y ) - ( x + y ) = ( x + y ) + ( - ( x + y ) )                 1.     Definition 1. Subtraction: p - q = p + (-q).

      2.     ( x + y ) + ( - ( x + y ) ) = 0                                         2.     Property IX Additive Inverse p + (-p) = 0.

      3.     ( x + ( -x ) ) + ( y + ( -y ) ) = 0 + 0 = 0                         3.     Property IX and Property VII. IDENTITY: p+0 = p

      4.     ( x + y ) + ( ( -x ) + ( -y ) ) = ( x + y ) + ( - ( x + y ) )       4.     Properties XIX LHS(s) = 0 so = each other.

      5.     ( -x ) + ( -y ) = - ( x + y )                                             5.     Properties XXII If p = q, then p - r = q - r.

      6.         - x - y = - ( x + y )                                                   6.     Definition 1. Subtraction: p - q = p + (-q).

      7.     - ( x + y ) = -x - y                                                        7.     SYMMETRIC: XIV. If p = q then q = p.



In reason # 4 LHS means left hand side of an equation.

Note: This demonstration may promote another property:

      Property XXXXIIa:       z - ( x + y ) = z - x - y

Would you like to demonstrate Property XXXXIIa. ?



Properties:     List #1 Link     List #2 Link     List #3 Link     List #4 Link     List #5 Link   and   Demonstrations

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