Prove : For all integers, m,n,p,q: (m - n) - (p - q) = (m + q) - (n + p) Given :

Question

Prove: For all integers, m,n,p,q: (m - n) - (p - q) = (m + q) - (n + p)
Given: Axiom 1.1. If m, n, and p are integers, then:
1.1. i. m+n=n+m
1.1. ii. (m+n)+p=m+(n+p)
1.1. iii. m*(n+p)=mn+mp
1.1. iv. mn=nm
1.1. v. (m*n)*p=m*(n*p)
Axiom 1.2. There exists an integer 0 such that for every integer, m, m+0=m
Axiom 1.3. There exists an integer 1 such that 1 is not 0 and whenever m is integer, m*1=m.
Axiom 1.4. For each integer, m, there exists an integer, denoted by -m, such that m+(-m)=0
Axiom 1.5. Let m, n, and p be integers. In m*n=m*p and m does not equal 0, then n=p.

Definition of Subtraction: m-n is defined to be m+(-n)

Explanation / Answer

(m - n) - (p - q) now using axiom 1.4 = m -n - p -(-q) = m - n - p + q now using axiom 1.1 iv = m + q - n - p = (m + q) - (n + p) prooved

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