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Step * of Lemma continuous-composition

∀I:Interval
  (iproper(I)
  ⇒ (∀J:Interval. ∀f:{x:ℝ| x ∈ I}  ⟶ {y:ℝ| y ∈ J} . ∀g:J ⟶ℝ.
        (f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ J ⇒ g[f[x]] continuous for x ∈ I)))
BY
{ (Auto
   THEN (InstLemma `continuous-implies-functional` [⌜I⌝;⌜f⌝]⋅ THENA Auto)
   THEN (InstLemma `continuous-implies-functional` [⌜J⌝;⌜g⌝]⋅ THENA Auto)
   THEN BLemma `function-is-continuous`
   THEN Auto) }

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Latex:

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Latex:
\mforall{}I:Interval
    (iproper(I)
    {}\mRightarrow{}  (\mforall{}J:Interval.  \mforall{}f:\{x:\mBbbR{}|  x  \mmember{}  I\}    {}\mrightarrow{}  \{y:\mBbbR{}|  y  \mmember{}  J\}  .  \mforall{}g:J  {}\mrightarrow{}\mBbbR{}.
                (f[x]  continuous  for  x  \mmember{}  I  {}\mRightarrow{}  g[x]  continuous  for  x  \mmember{}  J  {}\mRightarrow{}  g[f[x]]  continuous  for  x  \mmember{}  I)))


By

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Latex:
(Auto
  THEN  (InstLemma  `continuous-implies-functional`  [\mkleeneopen{}I\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (InstLemma  `continuous-implies-functional`  [\mkleeneopen{}J\mkleeneclose{};\mkleeneopen{}g\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  BLemma  `function-is-continuous`
  THEN  Auto)

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