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Proof step 31 from proof less-mult-div-01 (previous step)

0 < i0 * j + j0, 0 < j1, i1 < j, j0 < j, 0 < j, 0 < (j0 * i1) / j + i0 * i1, 0 ≤ (j0 * i1) / j + i0 * i1, 0 ≤ i0 * j + j0, i0 * j + j0 ≤ (j0 * i1) / j + i0 * i1, 0 ≤ i0, 0 ≤ j0, i0 = 0 ∨ 0 ≤ i1, j0 = 0 ∨ 0 ≤ i1 ⊦

User applied rule insert lemma

And obtained the following new goals

Proof step 32:
⊦ 0 < i ∧ i < j ∧ 0 < i1 ∧ i1 < j → (i * i1) / j < i
Proof step 33:
0 < (j0 * i1) / j + i0 * i1, 0 < j, j0 < j, i1 < j, 0 < j1, 0 < i0 * j + j0, 0 ≤ j0, 0 ≤ i0, i0 * j + j0 ≤ (j0 * i1) / j + i0 * i1, 0 ≤ i0 * j + j0, 0 ≤ (j0 * i1) / j + i0 * i1, 0 < j0 ∧ j0 < j ∧ 0 < i1 ∧ i1 < j → (j0 * i1) / j < j0, j0 = 0 ∨ 0 ≤ i1, i0 = 0 ∨ 0 ≤ i1 ⊦

Rule argument

spec: <toplevel> name: prod-div-less seq: ⊦ 0 < i ∧ i < j ∧ 0 < i1 ∧ i1 < j → (i * i1) / j < i subst: [i, j, i1] → [j0, j, i1]

Additional information

Used terms: j0, j, i1