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Step * of Lemma range_sup_functionality_wrt_subinterval

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∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I}  ⟶ ℝ.
  ∀J:{J:Interval| icompact(J)} . (J ⊆ I  ⇒ (sup{f[x] | x ∈ J} ≤ sup{f[x] | x ∈ I})) 
  supposing ∀x,y:{x:ℝ| x ∈ I} .  ((x = y) ⇒ (f[x] = f[y]))
BY
{ (Auto THEN BLemma  `range_sup-bound` THEN Auto THEN InstLemma `rleq-range_sup` [⌜I⌝;⌜f⌝;⌜x⌝]⋅ THEN Auto) }

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Latex:

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Latex:
No  Annotations
\mforall{}I:\{I:Interval|  icompact(I)\}  .  \mforall{}f:\{x:\mBbbR{}|  x  \mmember{}  I\}    {}\mrightarrow{}  \mBbbR{}.
    \mforall{}J:\{J:Interval|  icompact(J)\}  .  (J  \msubseteq{}  I    {}\mRightarrow{}  (sup\{f[x]  |  x  \mmember{}  J\}  \mleq{}  sup\{f[x]  |  x  \mmember{}  I\})) 
    supposing  \mforall{}x,y:\{x:\mBbbR{}|  x  \mmember{}  I\}  .    ((x  =  y)  {}\mRightarrow{}  (f[x]  =  f[y]))


By

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Latex:
(Auto
  THEN  BLemma    `range\_sup-bound`
  THEN  Auto
  THEN  InstLemma  `rleq-range\_sup`  [\mkleeneopen{}I\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{}]\mcdot{}
  THEN  Auto)

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