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Step * 1 1 of Lemma arctangent-rtan


1. ∀x:ℝ. (r0 < (r1 + x^2))
2. ∀x:{x:ℝ| x ∈ (-(π/2), π/2)} . (r0 < rcos(x))
3. ∀x:{x:ℝ| x ∈ (-(π/2), π/2)} . (r0 < rcos(x)^2)
⊢ d(arctangent(rtan(x)))/dx = λx.r1 on (-(π/2), π/2)
BY
{ ((InstLemma `chain-rule` [⌜(-(π/2), π/2)⌝;⌜(-∞, ∞)⌝;⌜λ2x.rtan(x)⌝;⌜λ2x.(r1/rcos(x)^2)⌝;⌜λ2x.arctangent(x)⌝;⌜λ2x.(r1/r1
                                                                                                              + x^2)⌝]⋅
    THENW Auto
    )
   THEN Try ((Intros THEN BLemma `rdiv_functionality` THEN Auto))
   THEN Auto) }

1
.....antecedent..... 
1. ∀x:ℝ. (r0 < (r1 + x^2))
2. ∀x:{x:ℝ| x ∈ (-(π/2), π/2)} . (r0 < rcos(x))
3. ∀x:{x:ℝ| x ∈ (-(π/2), π/2)} . (r0 < rcos(x)^2)
⊢ maps-compact((-(π/2), π/2);(-∞, ∞);x.rtan(x))

2
1. ∀x:ℝ. (r0 < (r1 + x^2))
2. ∀x:{x:ℝ| x ∈ (-(π/2), π/2)} . (r0 < rcos(x))
3. ∀x:{x:ℝ| x ∈ (-(π/2), π/2)} . (r0 < rcos(x)^2)
4. d(arctangent(rtan(x)))/dx = λx.(r1/r1 + rtan(x)^2) * (r1/rcos(x)^2) on (-(π/2), π/2)
⊢ d(arctangent(rtan(x)))/dx = λx.r1 on (-(π/2), π/2)

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Latex:

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Latex:

1.  \mforall{}x:\mBbbR{}.  (r0  <  (r1  +  x\^{}2))
2.  \mforall{}x:\{x:\mBbbR{}|  x  \mmember{}  (-(\mpi{}/2),  \mpi{}/2)\}  .  (r0  <  rcos(x))
3.  \mforall{}x:\{x:\mBbbR{}|  x  \mmember{}  (-(\mpi{}/2),  \mpi{}/2)\}  .  (r0  <  rcos(x)\^{}2)
\mvdash{}  d(arctangent(rtan(x)))/dx  =  \mlambda{}x.r1  on  (-(\mpi{}/2),  \mpi{}/2)


By

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Latex:
((InstLemma  `chain-rule`  [\mkleeneopen{}(-(\mpi{}/2),  \mpi{}/2)\mkleeneclose{};\mkleeneopen{}(-\minfty{},  \minfty{})\mkleeneclose{};\mkleeneopen{}\mlambda{}\msubtwo{}x.rtan(x)\mkleeneclose{};\mkleeneopen{}\mlambda{}\msubtwo{}x.(r1/rcos(x)\^{}2)\mkleeneclose{};
    \mkleeneopen{}\mlambda{}\msubtwo{}x.arctangent(x)\mkleeneclose{};\mkleeneopen{}\mlambda{}\msubtwo{}x.(r1/r1  +  x\^{}2)\mkleeneclose{}]\mcdot{}
    THENW  Auto
    )
  THEN  Try  ((Intros  THEN  BLemma  `rdiv\_functionality`  THEN  Auto))
  THEN  Auto)

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