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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">Z</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">f3</font>, <font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>











<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c6">f2</font> &amp; <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2 &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> holds <br/>( <font color="Maroon" title="c5">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 ) ) )
 ;<br/>

<a NAME="E2:30"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E3:30"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E4:30"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a>, <a class="txt" href="fdiff_7.html#E2:30"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<a NAME="E5:30"/><i><font color="Green" title="E22">A5</font></i>: 
( <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> holds <br/>( <font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) ) )
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E1:30"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E6:30"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
( <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E3:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="fdiff_7.html#T25" target="_self" title="FDIFF_7:th.25">Th25</a></i>;<br/>
<a NAME="E7:30"/><i><font color="Green" title="E24">A7</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c6">f2</font> &amp; <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> 2 &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> holds <br/>( <font color="Maroon" title="c5">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 ) ) )
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E1:30"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E8:30"/><b>then </b><i><font color="Green" title="E25">A8</font></i>: 
( <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> 2</span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E4:30"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="fdiff_7.html#T27" target="_self" title="FDIFF_7:th.27">Th27</a></i>;<br/>
<a NAME="E9:30"/><i><font color="Green" title="E26">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> 2</span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E27">A10</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ;<br/>

<a NAME="E2:30_1"/><b>then </b><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E1:30"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E3:30_1"/><b>then </b><i><font color="Green" title="E29">A12</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></i>;<br/>
<a NAME="E4:30_1"/><i><font color="Green" title="E30">A13</font></i>: 
<font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E1:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="fdiff_7.html#E1:30_1"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>
<a NAME="E5:30_1"/>
<font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E1:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="fdiff_7.html#E1:30_1"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>
<a NAME="E6:30_1"/><b>then </b><i><font color="Green" title="E31">A14</font></i>: 
1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:30_1"/>
<font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E1:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="fdiff_7.html#E1:30_1"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>
<a NAME="E8:30_1"/><b>then </b>
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></i>;<br/>
<a NAME="E9:30_1"/><b>then </b><i><font color="Green" title="E32">A15</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_7.html#E6:30_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/>
<a NAME="E10:30_1"/><b>then </b><i><font color="Green" title="E33">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a>, <a class="txt" href="fdiff_7.html#E3:30_1"><i><font color="Green" title="E29">A12</font></i></a>, <a class="txt" href="fdiff_7.html#E4:30_1"><i><font color="Green" title="E30">A13</font></i></a></i>;<br/>
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a>, <a class="txt" href="fdiff_7.html#E2:30_1"><i><font color="Green" title="E28">A11</font></i></a></i>
<br/>.= 
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E3:30_1"><i><font color="Green" title="E29">A12</font></i></a>, <a class="txt" href="fdiff_7.html#E4:30_1"><i><font color="Green" title="E30">A13</font></i></a>, <a class="txt" href="fdiff_7.html#E9:30_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="square_1.html#T97" title="SQUARE_1:th.97">SQUARE_1:97</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E10:30_1"><i><font color="Green" title="E33">A16</font></i></a>, <a class="ref" href="fdiff_7.html#T3" target="_self" title="FDIFF_7:th.3">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E2:30_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> 2</span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_7.html#T1" target="_self" title="FDIFF_7:th.1">Th1</a></i>
;<br/>
<b>hence </b><a NAME="E12:30_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> 2</span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:30"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:30_2"/><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E8:30"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="fdiff_7.html#E6:30"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="fdiff_7.html#E1:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T26" title="FDIFF_1:th.26">FDIFF_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E6:30"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="fdiff_7.html#E1:30_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E8:30"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="fdiff_7.html#E1:30_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E5:30"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="fdiff_7.html#E3:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="fdiff_7.html#T25" target="_self" title="FDIFF_7:th.25">Th25</a>, <a class="txt" href="fdiff_7.html#E1:30_2"><i><font color="Green" title="E27">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> 2</span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E7:30"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="fdiff_7.html#E4:30"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="fdiff_7.html#T27" target="_self" title="FDIFF_7:th.27">Th27</a>, <a class="txt" href="fdiff_7.html#E1:30_2"><i><font color="Green" title="E27">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> 2</span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_7.html#E9:30"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="fdiff_7.html#E1:30_2"><i><font color="Green" title="E27">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100" title="XCMPLX_1:th.100">XCMPLX_1:100</a></i>
<br/>.= 
<a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:30_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:30"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f3</font></span>)</span></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos6.html#K1" title="SIN_COS6:func.1">arcsin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="fdiff_7.html#E8:30"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="fdiff_7.html#E6:30"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="fdiff_7.html#E1:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T26" title="FDIFF_1:th.26">FDIFF_1:26</a></i>;<br/>


</div>
