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Step * 2 of Lemma log-from_wf


1. a : {a:ℝ| r0 < a} 
2. b : {b:ℝ| |b - rlog(a)| ≤ (r1/r(10))} 
3. rlog(a) = cauchy-limit(n.logseq(a;b;n);λk.(TERMOF{logseq-converges-ext:o, \\v:l} a b (2 * k)))
4. log-from(a;b) = cauchy-limit(n.logseq(a;b;n);λk.(TERMOF{logseq-converges-ext:o, \\v:l} a b (2 * k))) ∈ (ℕ+ ⟶ ℤ)
⊢ log-from(a;b) ∈ {x:ℝ| x = rlog(a)} 
BY
{ ((Assert cauchy-limit(n.logseq(a;b;n);λk.(TERMOF{logseq-converges-ext:o, \\v:l} a b (2 * k))) ∈ ℝ BY
          (MemCD THEN Try (BLemma `converges-cauchy-witness`) THEN Auto))
   THEN (Assert ⌜cauchy-limit(n.logseq(a;b;n);λk.(TERMOF{logseq-converges-ext:o, \\v:l} a b (2 * k)))
                 = log-from(a;b)
                 ∈ ℝ⌝⋅
         THENA (MemTypeHD (-1) THEN Auto)
         )
   THEN Thin 4
   THEN MemTypeCD
   THEN Auto) }

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Latex:

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Latex:

1.  a  :  \{a:\mBbbR{}|  r0  <  a\} 
2.  b  :  \{b:\mBbbR{}|  |b  -  rlog(a)|  \mleq{}  (r1/r(10))\} 
3.  rlog(a)  =  cauchy-limit(n.logseq(a;b;n);\mlambda{}k.(TERMOF\{logseq-converges-ext:o,  \mbackslash{}\mbackslash{}v:l\}  a  b  (2  *  k)))
4.  log-from(a;b)
=  cauchy-limit(n.logseq(a;b;n);\mlambda{}k.(TERMOF\{logseq-converges-ext:o,  \mbackslash{}\mbackslash{}v:l\}  a  b  (2  *  k)))
\mvdash{}  log-from(a;b)  \mmember{}  \{x:\mBbbR{}|  x  =  rlog(a)\} 


By

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Latex:
((Assert  cauchy-limit(n.logseq(a;b;n);\mlambda{}k.(TERMOF\{logseq-converges-ext:o,  \mbackslash{}\mbackslash{}v:l\}  a  b  (2  *  k)))  \mmember{}  \mBbbR{}  BY
                (MemCD  THEN  Try  (BLemma  `converges-cauchy-witness`)  THEN  Auto))
  THEN  (Assert  \mkleeneopen{}cauchy-limit(n.logseq(a;b;n);\mlambda{}k.(TERMOF\{logseq-converges-ext:o,  \mbackslash{}\mbackslash{}v:l\}  a  b  (2  *  k)))
                              =  log-from(a;b)\mkleeneclose{}\mcdot{}
              THENA  (MemTypeHD  (-1)  THEN  Auto)
              )
  THEN  Thin  4
  THEN  MemTypeCD
  THEN  Auto)

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