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Step * 1 of Lemma union-nat-missing-prop


1. s1 : nat-missing-type()@i
2. m : {m:ℤ| (-1) ≤ m} @i
3. s3 : {L:ℕ List| l-ordered(ℕ;x,y.x < y;L) ∧ (∀x∈L.x < m)} @i
4. x : ℕ@i
5. 0 ≤ m
⊢ ↑member-nat-missing(x;union-nat-missing-pos(s1;m;s3)) ⇐⇒ (↑member-nat-missing(x;s1)) ∨ (↑member-nat-missing(x;<m, s3>\000C))
BY
{ (RWO "union-nat-missing-pos-prop" 0 THEN Try (Complete (Auto)) THEN RWO "assert-member-nat-missing" 0 THEN Auto) }

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Latex:

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Latex:

1.  s1  :  nat-missing-type()@i
2.  m  :  \{m:\mBbbZ{}|  (-1)  \mleq{}  m\}  @i
3.  s3  :  \{L:\mBbbN{}  List|  l-ordered(\mBbbN{};x,y.x  <  y;L)  \mwedge{}  (\mforall{}x\mmember{}L.x  <  m)\}  @i
4.  x  :  \mBbbN{}@i
5.  0  \mleq{}  m
\mvdash{}  \muparrow{}member-nat-missing(x;union-nat-missing-pos(s1;m;s3))
\mLeftarrow{}{}\mRightarrow{}  (\muparrow{}member-nat-missing(x;s1))  \mvee{}  (\muparrow{}member-nat-missing(x;<m,  s3>))


By

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Latex:
(RWO  "union-nat-missing-pos-prop"  0
  THEN  Try  (Complete  (Auto))
  THEN  RWO  "assert-member-nat-missing"  0
  THEN  Auto)

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