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

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">x0</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>









<b>assume </b><a NAME="E1:38"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Olive">x</font> ) &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">f</font> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x0</font> )
 ;<br/>

<a NAME="E2:38"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra5.html#D4">INTEGRA5:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:38"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">h</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">h</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="rsspace.html#K3">R_id</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rsspace.html#K3">R_id</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 0;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K3">R_id</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:38"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">x</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1">PARTFUN1:sch 1</a>();<br/></i>
<a NAME="E8:38"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:38_1"/>
( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>] implies <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>] implies <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x</font>) ) )
 
<b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a>, <a class="ref" href="integra6.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:38_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="integra6.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a>, <a class="ref" href="partfun1.html#T25">PARTFUN1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as   <a href="fdiff_1.html#NM2">LINEAR</a> <b>by </b><i>, <a class="ref" href="fdiff_1.html#D4">FDIFF_1:def 4</a></i>;<br/>
<a NAME="E11:38"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">R</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E12:38"/>
for <font color="Olive">h</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Olive">h</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Olive">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">h</font> be  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<a NAME="E1:38_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">e</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e0</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>set </b><font color="Maroon">e</font> = <font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2;<br/>
<b>assume </b><a NAME="E1:38_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e0</font>
 ;<br/>

<a NAME="E2:38_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e0</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:38_2_1"/><i><font color="Green">E59</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">t</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">t</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2 ) )
 <b>by </b><i><a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="fcont_1.html#T3">FCONT_1:3</a></i>;<br/>
<b>consider </b><font color="Maroon">N</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E6:38_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rcomp_1.html#T42">RCOMP_1:42</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:38_2_1"/><i><font color="Green">E67</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/>
<b>set </b><font color="Maroon">r</font> =  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>;<br/>
<a NAME="E9:38_2_1"/><i><font color="Green">E68</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="integra6.html#T8" target="_self">Th8</a>, , <a class="ref" href="xreal_1.html#T217">XREAL_1:217</a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<a NAME="E10:38_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> &amp;  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2 &amp;  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2 )
 
<b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a>, <a class="ref" href="square_1.html#T38">SQUARE_1:38</a></i>;<br/>
<a NAME="E11:38_2_1"/><b>then </b>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E12:38_2_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E13:38_2_1"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E14:38_2_1"/>
( <font color="Maroon">h</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:38_2_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E17:38_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E18:38_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E19:38_2_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
;<br/>
<a NAME="E20:38_2_1"/><b>then </b>
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E21:38_2_1"/><b>then </b>
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E22:38_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, , <a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E23:38_2_1"/><b>then </b>
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K3">R_id</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rsspace.html#D3">RSSPACE:def 3</a></i>;<br/>
<a NAME="E24:38_2_1"/><b>then </b>
<font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="rsspace.html#K3">R_id</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rsspace.html#K3">R_id</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E25:38_2_1"/><b>then </b>
<font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rsspace.html#K3">R_id</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rsspace.html#D3">RSSPACE:def 3</a></i>;<br/>
<a NAME="E26:38_2_1"/><b>then </b>
<font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rsspace.html#D3">RSSPACE:def 3</a></i>;<br/>
<a NAME="E27:38_2_1"/><b>then </b>
<font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E28:38_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E30:38_2_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">h</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">h</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E31:38_2_1"/><i><font color="Green">E77</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E32:38_2_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E33:38_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">g</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 
<b>by </b><i>, <a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E34:38_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i><a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rfunct_1.html#T101">RFUNCT_1:101</a></i>;<br/>
<a NAME="E35:38_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 
<b>by </b><i>, <a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra6.html#E11:38"><i><font color="Green">E56</font></i></a>, </i>;<br/>
<a NAME="E36:38_2_1"/>
<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E37:38_2_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E38:38_2_1"/>
for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span><a href="integra5.html#K3">']</a></span> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:38_2_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span><a href="integra5.html#K3">']</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:38_2_1_1"/>
(  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a>, <a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/>
<a NAME="E4:38_2_1_1"/><b>then </b>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E5:38_2_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="integra6.html#E4:38_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="integra5.html#D4">INTEGRA5:def 4</a></i>;<br/>
<a NAME="E6:38_2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_2_1_1_1_1"/>
( <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> or  not <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:38_2_1_1_1_1_1"/>
<font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:38_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> &amp; <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:38_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="integra6.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E4:38_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
 ex <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> )
 ;<br/><a NAME="E5:38_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:38_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E7:38_2_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E8:38_2_1_1_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E9:38_2_1_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:38_2_1_1_1_1_2"/><i><font color="Green">E87</font></i>: 
not <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:38_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> &amp; <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:38_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="integra6.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E4:38_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
 ex <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> &amp; <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> )
 ;<br/><a NAME="E5:38_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_2_1_1_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:38_2_1_1_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:38_2_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T49">XREAL_1:49</a></i>;<br/><a NAME="E3:38_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:38_2_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T16">XREAL_1:16</a></i>;<br/><a NAME="E5:38_2_1_1_1_1_2_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E6:38_2_1_1_1_1_2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:38_2_1_1_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:38_2_1_1_1_1_2_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:38_2_1_1_1_1_2_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:38_2_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E8:38_2_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E9:38_2_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E10:38_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E11:38_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:38_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E13:38_2_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E14:38_2_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra6.html#E7:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="integra6.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E15:38_2_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:38_2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="txt" href="integra6.html#E7:38"><i><font color="Green">E36</font></i></a>, , <a class="txt" href="integra6.html#E2:38_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:38_2_1"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T1" target="_self">Th1</a>, <a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, , <a class="txt" href="integra6.html#E7:38"><i><font color="Green">E36</font></i></a>, <a class="txt" href="integra6.html#E1:38_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra6.html#E2:38_2_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E40:38_2_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/>
<a NAME="E41:38_2_1"/><b>then </b>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="seq_1.html#T6">SEQ_1:6</a></i>;<br/>
<a NAME="E42:38_2_1"/><b>then </b><i><font color="Green">E96</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/>
<a NAME="E43:38_2_1"/><i><font color="Green">E97</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>;<br/>
<a NAME="E44:38_2_1"/><i><font color="Green">E115</font></i>: 
 <a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra6.html#E7:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="integra6.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E45:38_2_1"/>
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra6.html#E7:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, </i>;<br/>
<a NAME="E46:38_2_1"/><b>then </b>
<font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra6.html#E7:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, , , <a class="txt" href="integra6.html#E11:38"><i><font color="Green">E56</font></i></a>, , </i>;<br/>
<a NAME="E47:38_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E4:38_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="integra6.html#T16" target="_self">Th16</a>, <a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/>
<a NAME="E48:38_2_1"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e0</font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="integra6.html#T16" target="_self">Th16</a>, , <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E49:38_2_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E50:38_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/>
<a NAME="E51:38_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>;<br/>
<a NAME="E52:38_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/>
<b>hence </b><a NAME="E53:38_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e0</font>
 <b>by </b><i><a class="ref" href="integra6.html#T7" target="_self">Th7</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:38_2"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>

<b>hence </b><a NAME="E3:38_2"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="fdiff_1.html#NM1">REST</a> <b>by </b><i><a class="ref" href="integra6.html#T6" target="_self">Th6</a>, <a class="ref" href="fdiff_1.html#D3">FDIFF_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">N</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E15:38"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rcomp_1.html#T42">RCOMP_1:42</a></i>;<br/>
<a NAME="E16:38"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E4:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a></i>;<br/>
<a NAME="E17:38"/><i><font color="Green">E119</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:38_3"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:38_3"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:38_3"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K3">R_id</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rsspace.html#D3">RSSPACE:def 3</a></i>;<br/>
<a NAME="E4:38_3"/>
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E5:38_3"/><b>then </b>
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K3">R_id</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="rsspace.html#D3">RSSPACE:def 3</a></i>;<br/>
<a NAME="E6:38_3"/><b>then </b>
<font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font>)
 
<b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a>, <a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E7:38_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:38"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">F</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="integra6.html#T7" target="_self">Th7</a>, <a class="ref" href="fdiff_1.html#D5">FDIFF_1:def 5</a></i>;<br/>

<a NAME="E19:38"/>
<font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> 1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E20:38"/>
 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">F</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="integra6.html#T7" target="_self">Th7</a>, <a class="ref" href="integra6.html#T8" target="_self">Th8</a>, <a class="ref" href="integra6.html#T9" target="_self">Th9</a>, <a class="ref" href="fdiff_1.html#D6">FDIFF_1:def 6</a></i>;<br/>


</div>
