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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a> ;<br/><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>assume </b><a NAME="E1:29"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font>;<br/>
<a NAME="E2:29"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r3</font> = <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font> as    <a href="subset_1.html#M1">Element</a> of  <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="E3:29_1"/>
for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">r3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font>
 
;<br/>
<b>hence </b><a NAME="E4:29_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:29"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a>  st <br/>for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="jgraph_2.html#T8" target="_self">Th8</a>);<br/></i>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E5:29"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">g0</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:29"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">r1</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="E1:29_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r1</font> as    <a href="subset_1.html#M1">Element</a> of  <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="E3:29_2"/>
<font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:29_2"/>
<font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:29"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>



<b>assume </b><a NAME="E1:29_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>consider </b><font color="Maroon">r0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:29_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">r0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="frechet.html#T8">FRECHET:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E6:29_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:29_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_3_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1"><b>case </b></a><a NAME="E1:29_3_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_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="E1:29_3_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:29_3_1_1_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">r4</font> = <font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font>;<br/><a NAME="E2:29_3_1_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>reconsider </b><font color="Maroon">G1</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E4:29_3_1_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E5:29_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:29_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E7:29_3_1_1_1_1_1_1"/>
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T46">JORDAN6:46</a></i>;<br/><b>then consider </b><font color="Maroon">W1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:29_3_1_1_1_1_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T6" target="_self">Th6</a>, , </i>;<br/><b>set </b><font color="Maroon">W</font> = <font color="Maroon">W1</font>;<br/><a NAME="E10:29_3_1_1_1_1_1_1"/>
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <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">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:29_3_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_3_1_1_1_1_1_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g0</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">g0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pz</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="jgraph_2.html#E7:29"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E5:29_3_1_1_1_1_1_1_1"/>
<font color="Maroon">pz</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E6:29_3_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">pz</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:29_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E7:29"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">aa1</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pz</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E9:29_3_1_1_1_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa1</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_2.html#E7:29"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">rx</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jgraph_2.html#E7:29"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E11:29_3_1_1_1_1_1_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font> &amp; <font color="Maroon">aa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E5:29"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font>
<b>by </b><i><a class="ref" href="jgraph_2.html#T9" target="_self">Th9</a>, , <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/>
<a NAME="E13:29_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">rx</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_2.html#E1:29_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jgraph_2.html#E4:29_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E14:29_3_1_1_1_1_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa1</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_2.html#E4:29_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font>
<b>by </b><i><a class="ref" href="jgraph_2.html#T9" target="_self">Th9</a>, , <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/>
<b>hence </b><a NAME="E16:29_3_1_1_1_1_1_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">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E1:29_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jgraph_2.html#E6:29_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jgraph_2.html#E7:29_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:29_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E12:29_3_1_1_1_1_1_1"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:29"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:29_3_1_1_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">r4</font> = <font color="Maroon">r0</font>;<br/><b>reconsider </b><font color="Maroon">G1</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E3:29_3_1_1_1_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E4:29_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E5:29_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:29_3_1_1_1_1_1_2"/>
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T46">JORDAN6:46</a></i>;<br/><b>then consider </b><font color="Maroon">W1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:29_3_1_1_1_1_1_2"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T6" target="_self">Th6</a>, , </i>;<br/><b>set </b><font color="Maroon">W</font> = <font color="Maroon">W1</font>;<br/><a NAME="E9:29_3_1_1_1_1_1_2"/>
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <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">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:29_3_1_1_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_3_1_1_1_1_1_2_1"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g0</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">g0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pz</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="jgraph_2.html#E5:29"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">aa1</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pz</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<i><font color="Green">E150</font></i>: <font color="Maroon">x</font> = 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa1</font>
<b>by </b><i>, <a class="txt" href="jgraph_2.html#E5:29"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
0
<b>by </b><i/>
;<br/>
<a NAME="E7:29_3_1_1_1_1_1_2_1"/>
( <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font> )
 
<b>by </b><i>, , , <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<b>hence </b><a NAME="E8:29_3_1_1_1_1_1_2_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">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E7:29"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:29_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E11:29_3_1_1_1_1_1_2"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:29_3_1_1_1"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_2"><b>case </b></a><a NAME="E1:29_3_1_1_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:29_3_1_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:29_3_1_1_2"/><i><font color="Green">E153</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">r4</font> = <font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span>;<br/><a NAME="E4:29_3_1_1_2"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E5:29_3_1_1_2"/><i><font color="Green">E155</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><b>reconsider </b><font color="Maroon">G1</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E7:29_3_1_1_2"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E8:29_3_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E9:29_3_1_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:29"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E10:29_3_1_1_2"/>
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T46">JORDAN6:46</a></i>;<br/><b>then consider </b><font color="Maroon">W1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E12:29_3_1_1_2"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T6" target="_self">Th6</a>, <a class="txt" href="jgraph_2.html#E7:29"><i><font color="Green">E43</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">W</font> = <font color="Maroon">W1</font>;<br/><a NAME="E13:29_3_1_1_2"/>
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <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">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:29_3_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_3_1_1_2_1"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g0</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">g0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pz</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="jgraph_2.html#E1:29_3"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E5:29_3_1_1_2_1"/>
<font color="Maroon">pz</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E6:29_3_1_1_2_1"/><b>then </b>
<font color="Maroon">pz</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:29_3_1_1_2_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E1:29_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">aa1</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pz</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E9:29_3_1_1_2_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa1</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_2.html#E1:29_3"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E10:29_3_1_1_2_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aa1</font> &amp; <font color="Maroon">aa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="jgraph_2.html#E4:29_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E11:29_3_1_1_2_1"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<i><font color="Green">E164</font></i>: <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font>
<b>by </b><i>, </i>
;<br/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E14:29_3_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa1</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_2.html#E7:29_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E15:29_3_1_1_2_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">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r0</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E6:29_3"><i><font color="Green">E47</font></i></a>, , , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:29_3_1_1_2"/><b>then </b>
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E15:29_3_1_1_2"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:29_3"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:29"/><b>then </b>
<font color="Maroon">g0</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:29"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font> ) &amp; <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i/>;<br/>


</div>
