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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Z</font> being  <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>   st <font color="Olive" title="b1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </span>)</span> &amp; <font color="Olive" title="b1">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> holds <br/>( <span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b1">Z</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b1">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> ) )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </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> implies ( <span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <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#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </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#K1" title="SEQ_1:func.1">.</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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:92"/><span class="lab"><font color="Green" title="E73">A1</font></span>: 
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:92"/><span class="lab"><font color="Green" title="E74">A2</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <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#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </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#K1" title="SEQ_1:func.1">.</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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> ) )</span><br/>

<a NAME="E3:92"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
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"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> implies <span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:92_1"/><span class="lab"><font color="Green" title="E76">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x</font></span><br/>

<a NAME="E2:92_1"/>
 <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="ref" href="sin_cos9.html#T82" target="_self" title="SIN_COS9:th.82">Th82</a></span>;<br/>
<a NAME="E3:92_1"/><span class="kw">then </span>
 <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_1"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:92_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_1.html#T3" title="TAYLOR_1:th.3">TAYLOR_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A5</font></span>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></span>;<br/>
<a NAME="E5:92"/>
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#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </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#K1" title="SEQ_1:func.1">.</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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:92_2"/><span class="lab"><font color="Green" title="E77">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E2:92_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A7</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E3:92_2"/>
 <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="ref" href="sin_cos9.html#T82" target="_self" title="SIN_COS9:th.82">Th82</a></span>;<br/>
<a NAME="E4:92_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A8</font></span>: 
 <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_2"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></span>;<br/>
<a NAME="E5:92_2"/><span class="lab"><font color="Green" title="E80">A9</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E1:92_2"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </span>)</span>,<font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="txt" href="#E1:92_2"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> ,<font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92_2"><span class="lab"><font color="Green" title="E79">A8</font></span></a>, <a class="ref" href="taylor_1.html#T3" title="TAYLOR_1:th.3">TAYLOR_1:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_2"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="txt" href="#E5:92_2"><span class="lab"><font color="Green" title="E80">A9</font></span></a>, <a class="ref" href="sin_cos9.html#T76" target="_self" title="SIN_COS9:th.76">Th76</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E7:92_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:92"/>
( <span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <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#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a> </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#K1" title="SEQ_1:func.1">.</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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</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"><a href="sin_cos9.html#K2" title="SIN_COS9:func.2">arccot</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
