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Prove: Let x,y be in R such that x < y. There exists a z in...

Prove: Let x,y be in R such that x < y. There exists a z in R such that x < z < y.

Given:
Axiom 8.1. For all x,y,z in R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that for all z in R, x + 0 = x.
Axiom 8.3. There exists a real number 1 such that 1 ≠ 0 and whenever x is in R, x*1 = x.
Axiom 8.4. For each x in R, there exists a real number, denoted by -x, such that x+(-x) = 0
Axiom 8.5. For each x in R-{0}, there exists a number, denoted by x-1, such that x*x-1 = 1

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