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Step * 1 1 3 of Lemma arcsine-rsin

.....antecedent..... 
1. ∀x:{x:ℝ| x ∈ (-(π/2), π/2)} . (rsin(x) ∈ (r(-1), r1))
2. ∀x:{x:ℝ| (-(π/2) < x) ∧ (x < π/2)} . ((r(-1) < rsin(x)) ∧ (rsin(x) < r1))
⊢ d(rsin(x))/dx = λx.rcos(x) on (-(π/2), π/2)
BY
{ ((InstLemma `derivative_functionality_wrt_subinterval` [⌜(-∞, ∞)⌝]⋅ THENM BHyp -1 )
   THEN Auto
   THEN D 0
   THEN Reduce 0
   THEN Auto) }

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Latex:

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Latex:
.....antecedent..... 
1.  \mforall{}x:\{x:\mBbbR{}|  x  \mmember{}  (-(\mpi{}/2),  \mpi{}/2)\}  .  (rsin(x)  \mmember{}  (r(-1),  r1))
2.  \mforall{}x:\{x:\mBbbR{}|  (-(\mpi{}/2)  <  x)  \mwedge{}  (x  <  \mpi{}/2)\}  .  ((r(-1)  <  rsin(x))  \mwedge{}  (rsin(x)  <  r1))
\mvdash{}  d(rsin(x))/dx  =  \mlambda{}x.rcos(x)  on  (-(\mpi{}/2),  \mpi{}/2)


By

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Latex:
((InstLemma  `derivative\_functionality\_wrt\_subinterval`  [\mkleeneopen{}(-\minfty{},  \minfty{})\mkleeneclose{}]\mcdot{}  THENM  BHyp  -1  )
  THEN  Auto
  THEN  D  0
  THEN  Reduce  0
  THEN  Auto)

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