<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <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>  ex <font color="Olive" title="b5">Z</font> being  <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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">C</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font> &amp; <font color="Olive" title="b5">Z</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b5">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b5">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c2">C</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">Z</font> ) &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <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>  ex <font color="Olive" title="b5">Z</font> being  <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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">C</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font> &amp; <font color="Olive" title="b5">Z</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b5">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b5">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c2">C</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">Z</font> ) implies <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_1_2"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <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>  ex <font color="Olive" title="b5">Z</font> being  <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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">C</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font> &amp; <font color="Olive" title="b5">Z</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b5">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b5">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c2">C</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">Z</font> )
 ; <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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <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/> for <font color="Olive" title="b5">Z</font> being  <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>   holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> or  not <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">C</font> or  not <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font> or  not <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font> or  not <font color="Olive" title="b5">Z</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  or  not <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">Z</font> or  not <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> or  not <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> or  not <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b5">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  or  not <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b5">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  or  not <font color="Maroon" title="c2">C</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or  not <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">Z</font> ) or <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:8_1_2"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <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>  ex <font color="Olive" title="b5">Z</font> being  <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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">C</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font> &amp; <font color="Olive" title="b5">Z</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b5">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b5">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c2">C</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">Z</font> )
 ; <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">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">b1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">y1</font> being   <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> , <font color="Maroon" title="c9">Z1</font> being   <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> <span class="kw"> such that </span><br/><a NAME="E4:8_1_2"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
( <font color="Maroon" title="c5">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">C</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">x1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">y1</font> &amp; <font color="Maroon" title="c9">Z1</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z1</font> &amp; <font color="Maroon" title="c7">x1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c9">Z1</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c9">Z1</font> &amp; <font color="Maroon" title="c7">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c9">Z1</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c8">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c9">Z1</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c2">C</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">Z1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">a2</font>, <font color="Maroon" title="c11">b2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c12">x2</font>, <font color="Maroon" title="c13">y2</font> being   <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> , <font color="Maroon" title="c14">Z2</font> being   <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> <span class="kw"> such that </span><br/><a NAME="E6:8_1_2"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
( <font color="Maroon" title="c10">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">b2</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">C</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">x2</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">y2</font> &amp; <font color="Maroon" title="c14">Z2</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">Z2</font> &amp; <font color="Maroon" title="c12">x2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c14">Z2</font> &amp; <font color="Maroon" title="c13">y2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c14">Z2</font> &amp; <font color="Maroon" title="c12">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c14">Z2</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c13">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c14">Z2</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c2">C</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c12">x2</font>,<font color="Maroon" title="c13">y2</font>,<font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c14">Z2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2"><span class="lab"><font color="Green" title="E8">A4</font></span></a></span>;<br/>
<a NAME="E7:8_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">C</font> is   <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></span>;<br/>
<a NAME="E8:8_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A8</font></span>: 
( <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a2</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">Z3</font> = <font color="Maroon" title="c9">Z1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">Z2</font> as   <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/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">ZZ1</font> = <font color="Maroon" title="c9">Z1</font>, <font color="Maroon" title="c17">ZZ2</font> = <font color="Maroon" title="c14">Z2</font>, <font color="Maroon" title="c18">ZZ3</font> = <font color="Maroon" title="c15">Z3</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<a NAME="E11:8_1_2"/>
( <font color="Maroon" title="c16">ZZ1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c17">ZZ2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="borsuk_5.html#T62" title="BORSUK_5:th.62">BORSUK_5:62</a></span>;<br/>
<a NAME="E12:8_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">ZZ1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">ZZ2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></span>;<br/>
<a NAME="E13:8_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon" title="c15">Z3</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T62" title="BORSUK_5:th.62">BORSUK_5:62</a></span>;<br/>
<a NAME="E14:8_1_2"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:8_1_2_1"/>
for <font color="Olive" title="b1">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Maroon" title="c7">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c19">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies <font color="Maroon" title="c7">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_1_2_1_1"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
<font color="Maroon" title="c19">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <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="c7">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span><br/>

<a NAME="E2:8_1_2_1_1"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E3:8_1_2_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">x1</font></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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E4:8_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A14</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <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"><font color="Maroon" title="c7">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E2:8_1_2_1_1"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<span class="lab"><font color="Green" title="E17">A15</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="txt" href="#E4:8_1_2_1_1"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
;<br/>
<a NAME="E6:8_1_2_1_1"/><span class="lab"><font color="Green" title="E18">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E7:8_1_2_1_1"/><span class="lab"><font color="Green" title="E19">A17</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">x2</font></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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E8:8_1_2_1_1"/><span class="lab"><font color="Green" title="E20">A18</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <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"><font color="Maroon" title="c12">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E6:8_1_2_1_1"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="txt" href="#E7:8_1_2_1_1"><span class="lab"><font color="Green" title="E19">A17</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E1:8_1_2_1_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E1:8_1_2_1_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="txt" href="#E8:8_1_2_1_1"><span class="lab"><font color="Green" title="E20">A18</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">y2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:8_1_2_1_1"/>
<font color="Maroon" title="c7">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E5:8_1_2_1_1"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="complex1.html#T163" title="COMPLEX1:th.163">COMPLEX1:163</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>
<span class="kw">hence </span><a NAME="E2:8_1_2_1"/>
<font color="Maroon" title="c7">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="funct_1.html#T165" title="FUNCT_1:th.165">FUNCT_1:165</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="E15:8_1_2"/><span class="lab"><font color="Green" title="E14">A19</font></span>: 
<font color="Maroon" title="c8">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c13">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:8_1_2_2"/>
for <font color="Olive" title="b1">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Maroon" title="c8">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c19">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies <font color="Maroon" title="c8">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_1_2_2_1"/><span class="lab"><font color="Green" title="E15">A20</font></span>: 
<font color="Maroon" title="c19">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <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="c8">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span><br/>

<a NAME="E2:8_1_2_2_1"/><span class="lab"><font color="Green" title="E16">A21</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E3:8_1_2_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">x1</font></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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E4:8_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A22</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <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"><font color="Maroon" title="c7">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E2:8_1_2_2_1"><span class="lab"><font color="Green" title="E16">A21</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<span class="lab"><font color="Green" title="E18">A23</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_2_1"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c8">y1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_2_1"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="txt" href="#E4:8_1_2_2_1"><span class="lab"><font color="Green" title="E17">A22</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
;<br/>
<a NAME="E6:8_1_2_2_1"/><span class="lab"><font color="Green" title="E19">A24</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E7:8_1_2_2_1"/><span class="lab"><font color="Green" title="E20">A25</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">x2</font></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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E8:8_1_2_2_1"/><span class="lab"><font color="Green" title="E21">A26</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <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"><font color="Maroon" title="c12">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E6:8_1_2_2_1"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="txt" href="#E7:8_1_2_2_1"><span class="lab"><font color="Green" title="E20">A25</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E1:8_1_2_2_1"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c13">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E1:8_1_2_2_1"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="txt" href="#E8:8_1_2_2_1"><span class="lab"><font color="Green" title="E21">A26</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">y2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:8_1_2_2_1"/>
<font color="Maroon" title="c8">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E5:8_1_2_2_1"><span class="lab"><font color="Green" title="E18">A23</font></span></a>, <a class="ref" href="complex1.html#T163" title="COMPLEX1:th.163">COMPLEX1:163</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>
<span class="kw">hence </span><a NAME="E2:8_1_2_2"/>
<font color="Maroon" title="c8">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c13">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="funct_1.html#T165" title="FUNCT_1:th.165">FUNCT_1:165</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="E16:8_1_2"/><span class="lab"><font color="Green" title="E15">A27</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Z3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E17:8_1_2"/><span class="lab"><font color="Green" title="E16">A28</font></span>: 
( <font color="Maroon" title="c7">x1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c15">Z3</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c15">Z3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E13:8_1_2"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></span>;<br/>
<a NAME="E18:8_1_2"/><span class="lab"><font color="Green" title="E17">A29</font></span>: 
( <font color="Maroon" title="c12">x2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c15">Z3</font> &amp; <font color="Maroon" title="c13">y2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c15">Z3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E13:8_1_2"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">s1</font> = 
 <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c15">Z3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E13:8_1_2"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E16:8_1_2"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="txt" href="integr1c.html#E4"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>
<br/>.= 
 <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c12">x2</font>,<font color="Maroon" title="c13">y2</font>,<font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c15">Z3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E8:8_1_2"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E13:8_1_2"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E14:8_1_2"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E15:8_1_2"><span class="lab"><font color="Green" title="E14">A19</font></span></a>, <a class="txt" href="#E17:8_1_2"><span class="lab"><font color="Green" title="E16">A28</font></span></a>, <a class="txt" href="#E18:8_1_2"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="txt" href="integr1c.html#E5"><span class="lab"><font color="Green" title="E5">Lm2</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E6:8_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E13:8_1_2"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E16:8_1_2"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="txt" href="integr1c.html#E4"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</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>
