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

<b>let </b><font color="Maroon">r</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">o</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>set </b><font color="Maroon">D2</font> =  <a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">S1</font> =  <a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">OK</font> = <span class="p1">(<span class="default"><a href="brouwer.html#K1" target="_self">DiffElems</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>
<a NAME="E2:22"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>
 
;<br/>
<a NAME="E3:22"/><i><font color="Green">E58</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T9">TOPREALB:9</a></i>;<br/>
<a NAME="E4:22"/><i><font color="Green">E62</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:22"/>
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T17">TOPREAL9:17</a></i>;<br/>
<a NAME="E6:22"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">s</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:22"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">s</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<i><font color="Green">E65</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="toprns_1.html#T24">TOPRNS_1:24</a></i>
;<br/>
<a NAME="E9:22"/><b>then </b>
<font color="Maroon">o</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="topreal9.html#T8">TOPREAL9:8</a></i>;<br/>
<a NAME="E10:22"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">s</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:22"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">s</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:22_2"/><i><font color="Green">E75</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T19">TOPREAL9:19</a></i>;<br/><a NAME="E3:22_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><b>hence </b><a NAME="E4:22_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:22"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">s</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="brouwer.html#K1" target="_self">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">OK</font> = <span class="p1">(<span class="default"><a href="brouwer.html#K1" target="_self">DiffElems</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">TD</font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font>;<br/>
<b>let </b><font color="Maroon">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E15:22"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="abian.html#NR4" target="_self">has_no_fixpoint</a> 
 ;<br/>

<b>set </b><font color="Maroon">g</font> =  <a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font>;<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> ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span> );<br/>
<a NAME="E16:22"/><i><font color="Green">E78</font></i>: 
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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:22_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">x2</font>
;<br/>

<b>thus </b><a NAME="E4:22_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">xo</font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E18:22"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">xo</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="pscomp_1.html#S1">PSCOMP_1:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span> );<br/>
<a NAME="E19:22"/><i><font color="Green">E90</font></i>: 
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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:22_4"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">x2</font>
;<br/>

<b>thus </b><a NAME="E4:22_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">yo</font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E21:22"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">yo</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="pscomp_1.html#S1">PSCOMP_1:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Olive">x2</font> <a href="euclid.html#K21">`1</a> </span>)</span> );<br/>
<a NAME="E22:22"/><i><font color="Green">E93</font></i>: 
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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:22_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">x2</font>
;<br/>

<b>thus </b><a NAME="E4:22_5"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">dx</font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E24:22"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Maroon">dx</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="pscomp_1.html#S1">PSCOMP_1:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">y1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Olive">y2</font> <a href="euclid.html#K22">`2</a> </span>)</span> );<br/>
<a NAME="E25:22"/><i><font color="Green">E96</font></i>: 
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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:22_6"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">x2</font>
;<br/>

<b>thus </b><a NAME="E4:22_6"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">dy</font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E27:22"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">y</font>,<font color="Maroon">dy</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="pscomp_1.html#S1">PSCOMP_1:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> <a href="euclid.html#K21">`1</a>  );<br/>
<a NAME="E28:22"/><i><font color="Green">E99</font></i>: 
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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:22_7"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<font color="Maroon">x2</font> <a href="euclid.html#K21">`1</a> 
;<br/>

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">x2</font>
;<br/>

<b>thus </b><a NAME="E4:22_7"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">x2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">fx2</font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E30:22"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Maroon">fx2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="pscomp_1.html#S1">PSCOMP_1:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> <a href="euclid.html#K22">`2</a>  );<br/>
<a NAME="E31:22"/><i><font color="Green">E102</font></i>: 
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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">x</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:22_8"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>take </b>
<font color="Maroon">x2</font> <a href="euclid.html#K22">`2</a> 
;<br/>

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">x2</font>
;<br/>

<b>thus </b><a NAME="E4:22_8"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">x2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">fy2</font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E33:22"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">x</font>,<font color="Maroon">fy2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="pscomp_1.html#S1">PSCOMP_1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">Dx</font> = <font color="Maroon">dx</font>, <font color="Maroon">Dy</font> = <font color="Maroon">dy</font>, <font color="Maroon">fX2</font> = <font color="Maroon">fx2</font>, <font color="Maroon">fY2</font> = <font color="Maroon">fy2</font>, <font color="Maroon">Xo</font> = <font color="Maroon">xo</font>, <font color="Maroon">Yo</font> = <font color="Maroon">yo</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E35:22"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Maroon">Xo</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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> 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">Xo</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="22_9"><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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_9"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">Xo</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>
 <b>and </b><br/><a NAME="E2:22_9"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:22_9"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E5:22_9"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">Xo</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="borsuk_5.html#T62">BORSUK_5:62</a></i>;<br/>
<b>consider </b><font color="Maroon">g</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="E8:22_9"/><i><font color="Green">E109</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E9:22_9"/><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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/>
<b>set </b><font color="Maroon">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_9"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:22_9_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:22_9_1"/><b>then </b>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:22_9_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
;<br/>

<a NAME="E13:22_9"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E14:22_9"/>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E15:22_9"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:22_9"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<a NAME="E17:22_9"/>
<font color="Maroon">W2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T66">PSCOMP_1:66</a></i>;<br/>
<b>hence </b><a NAME="E18:22_9"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon">b</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="E19:22_9"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xo</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E21:22_9"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E22:22_9"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">Xo</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">a2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E24:22_9"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E25:22_9"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">xo</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E26:22_9"/>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E28:22_9"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E29:22_9"/><i><font color="Green">E118</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> )
 ;<br/>
<a NAME="E30:22_9"/>
<font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E31:22_9"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E32:22_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E33:22_9"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:22"/><b>then </b>
<font color="Maroon">Xo</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">xo</font> = <font color="Maroon">xo</font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<a NAME="E38:22"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Maroon">Yo</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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> 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">Yo</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="22_10"><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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_10"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">Yo</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>
 <b>and </b><br/><a NAME="E2:22_10"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:22_10"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E5:22_10"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">Yo</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="borsuk_5.html#T62">BORSUK_5:62</a></i>;<br/>
<b>consider </b><font color="Maroon">g</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="E8:22_10"/><i><font color="Green">E181</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E9:22_10"/><i><font color="Green">E182</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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/>
<b>set </b><font color="Maroon">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_10"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:22_10_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:22_10_1"/><b>then </b>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:22_10_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
;<br/>

<a NAME="E13:22_10"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E14:22_10"/>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E15:22_10"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:22_10"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<a NAME="E17:22_10"/>
<font color="Maroon">W2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T68">PSCOMP_1:68</a></i>;<br/>
<b>hence </b><a NAME="E18:22_10"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon">b</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="E19:22_10"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Yo</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E21:22_10"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E22:22_10"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">Yo</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">a2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E24:22_10"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E25:22_10"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">yo</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E26:22_10"/>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E28:22_10"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E29:22_10"/><i><font color="Green">E189</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> )
 ;<br/>
<a NAME="E30:22_10"/>
<font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E31:22_10"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E32:22_10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E33:22_10"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:22"/><b>then </b>
<font color="Maroon">Yo</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">yo</font> = <font color="Maroon">yo</font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<a NAME="E41:22"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Maroon">Dx</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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> 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">Dx</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="22_11"><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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_11"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">Dx</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>
 <b>and </b><br/><a NAME="E2:22_11"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:22_11"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E5:22_11"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">dx</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="borsuk_5.html#T62">BORSUK_5:62</a></i>;<br/>
<b>consider </b><font color="Maroon">g</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="E8:22_11"/><i><font color="Green">E194</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E9:22_11"/><i><font color="Green">E195</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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/>
<b>set </b><font color="Maroon">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span> ;<br/>
<a NAME="E11:22_11"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:22_11_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:22_11_1"/><b>then </b>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:22_11_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E13:22_11"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:22_11_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:22_11_2"/><b>then </b>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:22_11_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
;<br/>

<a NAME="E15:22_11"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E16:22_11"/><i><font color="Green">E198</font></i>: 
0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E17:22_11"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E18:22_11"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:22_11"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E20:22_11"/>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E21:22_11"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E22:22_11"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<a NAME="E23:22_11"/>
( <font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">W2</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T66">PSCOMP_1:66</a></i>;<br/>
<b>hence </b><a NAME="E24:22_11"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon">b</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="E25:22_11"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Dx</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E27:22_11"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E28:22_11"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">Dx</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">a2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E30:22_11"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E31:22_11"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">dx</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E32:22_11"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E34:22_11"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E35:22_11"/><i><font color="Green">E206</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E36:22_11"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E37:22_11"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">a1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E38:22_11"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E39:22_11"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E40:22_11"/><b>then </b><i><font color="Green">E209</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<a NAME="E41:22_11"/>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E43:22_11"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E44:22_11"/><i><font color="Green">E211</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E45:22_11"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E46:22_11"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E47:22_11"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E48:22_11"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E49:22_11"/><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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<a NAME="E50:22_11"/><b>then </b><i><font color="Green">E214</font></i>: 
<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">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="real_1.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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E51:22_11"/>
 <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"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="real_1.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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T143">COMPLEX1:143</a></i>;<br/>
<a NAME="E52:22_11"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="real_1.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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/>
<a NAME="E53:22_11"/><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">x1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E3:22"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E54:22_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a> </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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i>, <a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<b>hence </b><a NAME="E55:22_11"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E42:22"/><b>then </b>
<font color="Maroon">Dx</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">dx</font> = <font color="Maroon">dx</font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<a NAME="E44:22"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Maroon">Dy</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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> 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">Dy</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="22_12"><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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_12"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">Dy</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>
 <b>and </b><br/><a NAME="E2:22_12"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:22_12"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E5:22_12"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">dy</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="borsuk_5.html#T62">BORSUK_5:62</a></i>;<br/>
<b>consider </b><font color="Maroon">g</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="E8:22_12"/><i><font color="Green">E219</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E9:22_12"/><i><font color="Green">E220</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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/>
<b>set </b><font color="Maroon">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span> ;<br/>
<a NAME="E11:22_12"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:22_12_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:22_12_1"/><b>then </b>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:22_12_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E13:22_12"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:22_12_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:22_12_2"/><b>then </b>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:22_12_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
;<br/>

<a NAME="E15:22_12"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E16:22_12"/><i><font color="Green">E222</font></i>: 
0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E17:22_12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E18:22_12"/><b>then </b><i><font color="Green">E223</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:22_12"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E20:22_12"/>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E21:22_12"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E22:22_12"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<a NAME="E23:22_12"/>
( <font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">W2</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T68">PSCOMP_1:68</a></i>;<br/>
<b>hence </b><a NAME="E24:22_12"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon">b</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="E25:22_12"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Dy</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E27:22_12"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<font color="Maroon">W2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E28:22_12"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">Dy</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">a2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E30:22_12"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E31:22_12"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">dy</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E32:22_12"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E34:22_12"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E35:22_12"/><i><font color="Green">E230</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font>
 <b>and </b><br/><a NAME="E36:22_12"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E37:22_12"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">a1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E38:22_12"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E39:22_12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E40:22_12"/><b>then </b><i><font color="Green">E233</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<a NAME="E41:22_12"/>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E43:22_12"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E44:22_12"/><i><font color="Green">E235</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y2</font>
 <b>and </b><br/><a NAME="E45:22_12"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">y2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/>
<a NAME="E46:22_12"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E47:22_12"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y2</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E48:22_12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E49:22_12"/><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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<a NAME="E50:22_12"/><b>then </b><i><font color="Green">E238</font></i>: 
<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">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font></span>)</span></span>)</span> <a href="real_1.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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E51:22_12"/>
 <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"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font></span>)</span></span>)</span> <a href="real_1.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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T143">COMPLEX1:143</a></i>;<br/>
<a NAME="E52:22_12"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font></span>)</span></span>)</span> <a href="real_1.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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/>
<a NAME="E53:22_12"/><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">y1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E3:22"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E54:22_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a> </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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i>, <a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<b>hence </b><a NAME="E55:22_12"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:22"/><b>then </b>
<font color="Maroon">Dy</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">dy</font> = <font color="Maroon">dy</font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<a NAME="E47:22"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Maroon">fX2</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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> 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">fX2</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="22_13"><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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_13"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">fX2</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>
 <b>and </b><br/><a NAME="E2:22_13"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:22_13"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E5:22_13"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">fX2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="borsuk_5.html#T62">BORSUK_5:62</a></i>;<br/>
<b>consider </b><font color="Maroon">g</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="E8:22_13"/><i><font color="Green">E243</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E9:22_13"/><i><font color="Green">E244</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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/>
<b>set </b><font color="Maroon">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_13"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:22_13_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:22_13_1"/><b>then </b>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:22_13_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W1</font></span><a href="borsuk_1.html#K6">:]</a></span>
;<br/>

<a NAME="E13:22_13"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E14:22_13"/>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E15:22_13"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:22_13"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<a NAME="E17:22_13"/>
<font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T66">PSCOMP_1:66</a></i>;<br/>
<b>hence </b><a NAME="E18:22_13"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W1</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon">b</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="E19:22_13"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fX2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E21:22_13"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E22:22_13"/><i><font color="Green">E247</font></i>: 
<font color="Maroon">fX2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">a2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E24:22_13"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E25:22_13"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">fX2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i/>;<br/>
<a NAME="E26:22_13"/>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E28:22_13"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E29:22_13"/><i><font color="Green">E251</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E30:22_13"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font>
 ;<br/>
<a NAME="E31:22_13"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E32:22_13"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E33:22_13"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E34:22_13"/><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">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<a NAME="E35:22_13"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/>
<a NAME="E36:22_13"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 
;<br/>
<a NAME="E37:22_13"/><b>then </b>
<font color="Maroon">a2</font> <a href="euclid.html#K21">`1</a>  <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"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></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/>
<b>hence </b><a NAME="E38:22_13"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:22"/><b>then </b>
<font color="Maroon">fX2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fx2</font> = <font color="Maroon">fx2</font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<a NAME="E50:22"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Maroon">fY2</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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> 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">fY2</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="22_14"><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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_14"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">fY2</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>
 <b>and </b><br/><a NAME="E2:22_14"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:22_14"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E5:22_14"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">fY2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="borsuk_5.html#T62">BORSUK_5:62</a></i>;<br/>
<b>consider </b><font color="Maroon">g</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="E8:22_14"/><i><font color="Green">E258</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E9:22_14"/><i><font color="Green">E259</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T40">RCOMP_1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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/>
<b>set </b><font color="Maroon">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span> ;<br/>
<a NAME="E11:22_14"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:22_14_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:22_14_1"/><b>then </b>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:22_14_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">y</font> is    <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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W1</font></span><a href="borsuk_1.html#K6">:]</a></span>
;<br/>

<a NAME="E13:22_14"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E14:22_14"/>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E15:22_14"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:22_14"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<a NAME="E17:22_14"/>
<font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T68">PSCOMP_1:68</a></i>;<br/>
<b>hence </b><a NAME="E18:22_14"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W1</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon">b</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="E19:22_14"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fY2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E21:22_14"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>,<font color="Maroon">W1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E22:22_14"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">fY2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">a2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E24:22_14"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E25:22_14"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">fY2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i/>;<br/>
<a NAME="E26:22_14"/>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E28:22_14"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E29:22_14"/><i><font color="Green">E266</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font>
 <b>and </b><br/><a NAME="E30:22_14"/><i><font color="Green">E267</font></i>: 
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font>
 ;<br/>
<a NAME="E31:22_14"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E32:22_14"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E33:22_14"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E34:22_14"/><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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<a NAME="E35:22_14"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/>
<a NAME="E36:22_14"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 
;<br/>
<a NAME="E37:22_14"/><b>then </b>
<font color="Maroon">a2</font> <a href="euclid.html#K22">`2</a>  <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"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></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/>
<b>hence </b><a NAME="E38:22_14"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E51:22"/><b>then </b>
<font color="Maroon">fY2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fy2</font> = <font color="Maroon">fy2</font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rr</font> = <font color="Maroon">r</font> <a href="square_1.html#K3">^2</a>  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/>
<b>set </b><font color="Maroon">f1</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">rr</font>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">rr</font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">Zf1</font> = <font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font>;<br/>
<b>set </b><font color="Maroon">Zfx2</font> = <font color="Maroon">fx2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font>;<br/>
<b>set </b><font color="Maroon">Zfy2</font> = <font color="Maroon">fy2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font>;<br/>
<b>set </b><font color="Maroon">Zdx</font> = <font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font>;<br/>
<b>set </b><font color="Maroon">Zdy</font> = <font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font>;<br/>
<b>set </b><font color="Maroon">Zxo</font> = <font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font>;<br/>
<b>set </b><font color="Maroon">Zyo</font> = <font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font>;<br/>
<b>set </b><font color="Maroon">xx</font> = <span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">yy</font> = <span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span>;<br/>
<a NAME="E55:22"/><i><font color="Green">E279</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">OK</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E56:22"/><i><font color="Green">E280</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E57:22"/><i><font color="Green">E281</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E58:22"/><i><font color="Green">E282</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E59:22"/><i><font color="Green">E283</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>, <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_15"/><i><font color="Green">E284</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:22_15"/>
( <font color="Maroon">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon">z</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E3:22_15"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="brouwer.html#K1" target="_self">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:22_15"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E60:22"/><i><font color="Green">E285</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">dx</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>, <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_16"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:22_16"/><b>then </b><i><font color="Green">E286</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:22_16"/>
<span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">dx</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E61:22"/><i><font color="Green">E287</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">dy</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>, <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_17"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:22_17"/><b>then </b><i><font color="Green">E288</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:22_17"/>
<span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">dy</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E62:22"/><i><font color="Green">E289</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">fx2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">fx2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>, <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_18"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:22_18"/><b>then </b><i><font color="Green">E290</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:22_18"/>
<span class="p1">(<span class="default"><font color="Maroon">fx2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">fx2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E63:22"/><i><font color="Green">E291</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">fy2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">fy2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>, <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_19"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:22_19"/><b>then </b><i><font color="Green">E292</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:22_19"/>
<span class="p1">(<span class="default"><font color="Maroon">fy2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">fy2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E64:22"/><i><font color="Green">E293</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>, <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_20"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:22_20"/><b>then </b><i><font color="Green">E294</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:22_20"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E65:22"/><i><font color="Green">E295</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">xo</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>, <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_21"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:22_21"/><b>then </b><i><font color="Green">E296</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:22_21"/>
<span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">xo</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E66:22"/><i><font color="Green">E297</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">yo</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_22"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>, <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_22"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:22_22"/><b>then </b><i><font color="Green">E298</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:22_22"/>
<span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">yo</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_23"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:22_23"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_23"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:22_23"/><i><font color="Green">E300</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:22_23"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="brouwer.html#K1" target="_self">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">z</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:22_23"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E8:22_23"/><i><font color="Green">E302</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E9:22_23"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>, <font color="Maroon">a2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span><b> such that </b><br/><a NAME="E11:22_23"/><i><font color="Green">E303</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/><a NAME="E12:22_23"/><i><font color="Green">E304</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>consider </b><font color="Maroon">x3</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x4</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E14:22_23"/><i><font color="Green">E307</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E15:22_23"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">dx</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x4</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E16:22_23"/><i><font color="Green">E309</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>consider </b><font color="Maroon">x5</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x6</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E18:22_23"/><i><font color="Green">E312</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x5</font>,<font color="Maroon">x6</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E19:22_23"/><i><font color="Green">E313</font></i>: 
<font color="Maroon">dy</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x5</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x6</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E20:22_23"/><i><font color="Green">E314</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x5</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x6</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>set </b><font color="Maroon">r1</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>;<br/><b>set </b><font color="Maroon">r2</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/><a NAME="E21:22_23"/><i><font color="Green">E315</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">dx</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E22:22_23"/><i><font color="Green">E316</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">dy</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, , </i>;<br/><i><font color="Green">E317</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i>, , , <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, <a class="txt" href="brouwer.html#E3:22"><i><font color="Green">E58</font></i></a>, <a class="ref" href="brouwer.html#T3" target="_self">Th3</a>, <a class="ref" href="brouwer.html#T4" target="_self">Th4</a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/><a NAME="E24:22_23"/><i><font color="Green">E318</font></i>: 
0 <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_23_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_23_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:22_23_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T2">COMPLEX1:2</a></i>;<br/><a NAME="E3:22_23_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">y</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><b>hence </b><a NAME="E4:22_23_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E26:22_23"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="txt" href="brouwer.html#E4:22"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> as  <a href="pscomp_1.html#V9">continuous</a> <a href="partfun3.html#V1">positive-yielding</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span> <b>by </b><i><a class="ref" href="partfun3.html#D1">PARTFUN3:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">p1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">p2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span>;<br/>
<a NAME="E69:22"/><i><font color="Green">E319</font></i>: 
 <a href="relset_1.html#K4">dom</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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E70:22"/><i><font color="Green">E320</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E71:22"/><i><font color="Green">E321</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E72:22"/><i><font color="Green">E322</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_24"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:22_24"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_24"/><i><font color="Green">E323</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:22_24"/><i><font color="Green">E324</font></i>: 
<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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:22_24"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="brouwer.html#K1" target="_self">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">z</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:22_24"/><i><font color="Green">E325</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E8:22_24"/><i><font color="Green">E326</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/><b>set </b><font color="Maroon">r3</font> = <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>;<br/><b>set </b><font color="Maroon">r4</font> = <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/><a NAME="E9:22_24"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>, <font color="Maroon">a2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span><b> such that </b><br/><a NAME="E11:22_24"/><i><font color="Green">E329</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/><a NAME="E12:22_24"/><i><font color="Green">E330</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E13:22_24"/><b>then </b><i><font color="Green">E331</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, </i>;<br/><a NAME="E14:22_24"/><i><font color="Green">E332</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">xo</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , </i>;<br/><a NAME="E15:22_24"/><i><font color="Green">E333</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">yo</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , </i>;<br/><b>consider </b><font color="Maroon">x9</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x10</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E17:22_24"/><i><font color="Green">E336</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x9</font>,<font color="Maroon">x10</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E18:22_24"/><i><font color="Green">E337</font></i>: 
<font color="Maroon">xo</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x10</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E19:22_24"/><i><font color="Green">E338</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x9</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x10</font> )
 <b>by </b><i><a class="ref" href="brouwer.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>consider </b><font color="Maroon">x11</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x12</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E21:22_24"/><i><font color="Green">E341</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x11</font>,<font color="Maroon">x12</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E22:22_24"/><i><font color="Green">E342</font></i>: 
<font color="Maroon">yo</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x12</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E23:22_24"/><i><font color="Green">E343</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x11</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x12</font> )
 <b>by </b><i><a class="txt" href="brouwer.html#E6:22"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><i><font color="Green">E344</font></i>: <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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i>, , , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i>, , , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i>, , , <a class="txt" href="brouwer.html#E3:22"><i><font color="Green">E58</font></i></a>, <a class="ref" href="brouwer.html#T3" target="_self">Th3</a>, , <a class="txt" href="brouwer.html#E4:22"><i><font color="Green">E62</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a>, <a class="txt" href="brouwer.html#E7:22"><i><font color="Green">E64</font></i></a>, <a class="txt" href="brouwer.html#E8:22"><i><font color="Green">E65</font></i></a></i>
;<br/><a NAME="E25:22_24"/><i><font color="Green">E345</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/><a NAME="E26:22_24"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E27:22_24"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="topreal9.html#T8">TOPREAL9:8</a></i>;<br/><a NAME="E28:22_24"/><b>then </b><i><font color="Green">E346</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_1.html#T46">JGRAPH_1:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T8">TOPREAL3:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T8">TOPREAL3:8</a></i>
;<br/><a NAME="E30:22_24"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E11:22"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E31:22_24"/>
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> as  <a href="pscomp_1.html#V9">continuous</a> <a href="partfun3.html#V3">nonpositive-yielding</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span> <b>by </b><i><a class="ref" href="partfun3.html#D3">PARTFUN3:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">pp</font> = <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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">x3</font> = <span class="p1">(<span class="default"><font color="Maroon">fx2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">y3</font> = <span class="p1">(<span class="default"><font color="Maroon">fy2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">X3</font> = <span class="p1">(<span class="default"><font color="Maroon">fx2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span>, <font color="Maroon">Y3</font> = <span class="p1">(<span class="default"><font color="Maroon">fy2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">X3</font>,<font color="Maroon">Y3</font></span><a href="funct_3.html#K14">:&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="topreala.html#K2">R2Homeomorphism</a> ;<br/>
<a NAME="E76:22"/><i><font color="Green">E352</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <span class="p2"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">X3</font>,<font color="Maroon">Y3</font></span><a href="funct_3.html#K14">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_25"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<a NAME="E1:22_25"/><i><font color="Green">E353</font></i>: 
not <font color="Maroon">x</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="abian.html#D5">ABIAN:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">z</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:22_25"/><i><font color="Green">E354</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:22_25"/><i><font color="Green">E355</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:22_25"/><i><font color="Green">E356</font></i>: 
 <a href="brouwer.html#K4" target="_self">HC</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="brouwer.html#K3" target="_self">HC</a> <font color="Maroon">z</font>,<font color="Maroon">y</font>,<font color="Maroon">o</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:22_25"/><i><font color="Green">E357</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="abian.html#D4">ABIAN:def 4</a></i>;<br/>
<a NAME="E7:22_25"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="brouwer.html#K1" target="_self">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i>, <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E8:22_25"/><b>then </b><i><font color="Green">E358</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<b>by </b><i>, <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">r1</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">r2</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">r3</font> = <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">r4</font> = <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E10:22_25"/><i><font color="Green">E359</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E11:22_25"/><i><font color="Green">E360</font></i>: 
<font color="Maroon">fx2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i/>;<br/>
<a NAME="E12:22_25"/><i><font color="Green">E361</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E3:22"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">x7</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x8</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E14:22_25"/><i><font color="Green">E364</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x7</font>,<font color="Maroon">x8</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E15:22_25"/><i><font color="Green">E365</font></i>: 
<font color="Maroon">fy2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x8</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i/>;<br/>
<a NAME="E16:22_25"/><i><font color="Green">E366</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x8</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">x3</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x4</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E18:22_25"/><i><font color="Green">E367</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E19:22_25"/><i><font color="Green">E368</font></i>: 
<font color="Maroon">dx</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x4</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E20:22_25"/><i><font color="Green">E369</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> )
 
<b>by </b><i><a class="txt" href="brouwer.html#E7:22"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">x5</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x6</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E22:22_25"/><i><font color="Green">E370</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x5</font>,<font color="Maroon">x6</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E23:22_25"/><i><font color="Green">E371</font></i>: 
<font color="Maroon">dy</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x5</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x6</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E24:22_25"/><i><font color="Green">E372</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x5</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x6</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">x9</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x10</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E26:22_25"/><i><font color="Green">E373</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x9</font>,<font color="Maroon">x10</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E27:22_25"/><i><font color="Green">E374</font></i>: 
<font color="Maroon">xo</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x10</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E28:22_25"/><i><font color="Green">E375</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x9</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x10</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">x11</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">x12</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E30:22_25"/><i><font color="Green">E376</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x11</font>,<font color="Maroon">x12</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E31:22_25"/><i><font color="Green">E377</font></i>: 
<font color="Maroon">yo</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x12</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E32:22_25"/><i><font color="Green">E378</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x11</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x12</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E33:22_25"/><i><font color="Green">E379</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">X3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E34:22_25"/><i><font color="Green">E380</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Y3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E35:22_25"/><i><font color="Green">E381</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E36:22_25"/><i><font color="Green">E382</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E37:22_25"/><i><font color="Green">E383</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E38:22_25"/><i><font color="Green">E384</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E39:22_25"/><i><font color="Green">E385</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E40:22_25"/><i><font color="Green">E386</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E41:22_25"/><i><font color="Green">E387</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E42:22_25"/><i><font color="Green">E388</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E43:22_25"/><i><font color="Green">E389</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E44:22_25"/><i><font color="Green">E390</font></i>: 
 <a href="relset_1.html#K4">dom</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"><span class="p5">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E45:22_25"/><i><font color="Green">E391</font></i>: 
 <a href="relset_1.html#K4">dom</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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E46:22_25"/><i><font color="Green">E392</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E47:22_25"/><i><font color="Green">E393</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">dx</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i>, <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , </i>;<br/>
<a NAME="E48:22_25"/><i><font color="Green">E394</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">dy</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i>, <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , </i>;<br/>
<a NAME="E49:22_25"/><i><font color="Green">E395</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i>, <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , </i>;<br/>
<a NAME="E50:22_25"/><i><font color="Green">E396</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">xo</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i>, <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , </i>;<br/>
<a NAME="E51:22_25"/><i><font color="Green">E397</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">yo</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i>, <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , </i>;<br/>
<i><font color="Green">E398</font></i>: <font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , <a class="txt" href="brouwer.html#E18:22"><i><font color="Green">E80</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, <a class="txt" href="brouwer.html#E8:22"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="brouwer.html#T6" target="_self">Th6</a>, <a class="txt" href="brouwer.html#E19:22"><i><font color="Green">E90</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a>, , <a class="txt" href="brouwer.html#E11:22"><i><font color="Green">E68</font></i></a>, <a class="ref" href="brouwer.html#T5" target="_self">Th5</a></i>
;<br/>
<a NAME="E53:22_25"/><i><font color="Green">E399</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E54:22_25"/><i><font color="Green">E400</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E55:22_25"/><i><font color="Green">E401</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E56:22_25"/><i><font color="Green">E402</font></i>: 
<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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</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">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, <a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a>, <a class="txt" href="brouwer.html#E19:22"><i><font color="Green">E90</font></i></a>, <a class="txt" href="brouwer.html#E8:22"><i><font color="Green">E65</font></i></a>, , <a class="txt" href="brouwer.html#E22:22"><i><font color="Green">E93</font></i></a>, <a class="ref" href="brouwer.html#D3" target="_self">Def3</a>, , <a class="txt" href="brouwer.html#E27:22"><i><font color="Green">E98</font></i></a>, <a class="txt" href="brouwer.html#E25:22"><i><font color="Green">E96</font></i></a>, , , <a class="txt" href="brouwer.html#E11:22"><i><font color="Green">E68</font></i></a>, <a class="ref" href="brouwer.html#T5" target="_self">Th5</a>, , , </i>;<br/>
<i><font color="Green">E403</font></i>: <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , , <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E21:22"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i>, , <a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, <a class="ref" href="brouwer.html#D3" target="_self">Def3</a>, , <a class="txt" href="brouwer.html#E22:22"><i><font color="Green">E93</font></i></a>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a>, , </i>
;<br/>
<a NAME="E58:22_25"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E404</font></i>: <span class="p1">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> = 
 <a href="square_1.html#K6">sqrt</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , <a class="ref" href="partfun3.html#D5">PARTFUN3:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K6">sqrt</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , , <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
 <a href="square_1.html#K6">sqrt</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, <a class="txt" href="brouwer.html#E24:22"><i><font color="Green">E95</font></i></a>, <a class="txt" href="brouwer.html#E28:22"><i><font color="Green">E99</font></i></a>, , , , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<a NAME="E60:22_25"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E405</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , <a class="ref" href="rfunct_1.html#D4">RFUNCT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , , <a class="txt" href="brouwer.html#E24:22"><i><font color="Green">E95</font></i></a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , <a class="txt" href="brouwer.html#E19:22"><i><font color="Green">E90</font></i></a>, <a class="txt" href="brouwer.html#E8:22"><i><font color="Green">E65</font></i></a>, , <a class="txt" href="brouwer.html#E22:22"><i><font color="Green">E93</font></i></a>, <a class="ref" href="brouwer.html#D3" target="_self">Def3</a>, , <a class="txt" href="brouwer.html#E25:22"><i><font color="Green">E96</font></i></a>, , <a class="txt" href="brouwer.html#E11:22"><i><font color="Green">E68</font></i></a>, <a class="ref" href="brouwer.html#T5" target="_self">Th5</a>, , , , , <a class="txt" href="brouwer.html#E27:22"><i><font color="Green">E98</font></i></a>, <a class="txt" href="brouwer.html#E30:22"><i><font color="Green">E101</font></i></a>, , <a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
;<br/>
<i><font color="Green">E406</font></i>: <font color="Maroon">X3</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fx2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</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"><span class="p5">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p0">(<span class="default"><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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , <a class="txt" href="brouwer.html#E15:22"><i><font color="Green">E76</font></i></a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</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"><span class="p5">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p0">(<span class="default"><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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , , <a class="ref" href="brouwer.html#T3" target="_self">Th3</a>, <a class="ref" href="brouwer.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</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"><span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, <a class="txt" href="brouwer.html#E19:22"><i><font color="Green">E90</font></i></a>, <a class="txt" href="brouwer.html#E8:22"><i><font color="Green">E65</font></i></a>, , <a class="txt" href="brouwer.html#E31:22"><i><font color="Green">E102</font></i></a>, , <a class="txt" href="brouwer.html#E16:22"><i><font color="Green">E78</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<i><font color="Green">E407</font></i>: <font color="Maroon">Y3</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fy2</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</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"><span class="p5">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p0">(<span class="default"><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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</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"><span class="p5">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p0">(<span class="default"><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">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">xo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dx</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">yo</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">dy</font> <a href="pscomp_1.html#K12">||</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , , <a class="txt" href="brouwer.html#E4:22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="brouwer.html#E6:22"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</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"><span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , <a class="txt" href="brouwer.html#E11:22"><i><font color="Green">E68</font></i></a>, <a class="ref" href="brouwer.html#T5" target="_self">Th5</a>, <a class="txt" href="brouwer.html#E31:22"><i><font color="Green">E102</font></i></a>, , , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
 <a href="brouwer.html#K4" target="_self">HC</a> <font color="Maroon">x</font>,<font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</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">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K6">sqrt</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</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">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K6">sqrt</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , , </i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X3</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Y3</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i>, <a class="txt" href="brouwer.html#E33:22"><i><font color="Green">E104</font></i></a>, <a class="ref" href="euclid.html#D16">EUCLID:def 16</a></i>
<br/>.= 
<a href="topreala.html#K2">R2Homeomorphism</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X3</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Y3</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="topreala.html#D2">TOPREALA:def 2</a></i>
<br/>.= 
<a href="topreala.html#K2">R2Homeomorphism</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">X3</font>,<font color="Maroon">Y3</font></span><a href="funct_3.html#K14">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, <a class="txt" href="brouwer.html#E15:22"><i><font color="Green">E76</font></i></a>, , , <a class="ref" href="funct_3.html#T69">FUNCT_3:69</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <span class="p2"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">X3</font>,<font color="Maroon">Y3</font></span><a href="funct_3.html#K14">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
<b>by </b><i><a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, , <a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E77:22"/>
( <font color="Maroon">X3</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">Y3</font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">X3</font>,<font color="Maroon">Y3</font></span><a href="funct_3.html#K14">:&gt;</a></span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topalg_2.html#K3">R^1</a> ,<a href="topalg_2.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="yellow12.html#T41">YELLOW12:41</a></i>;<br/>
<a NAME="E79:22"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>  st <span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span> 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; <span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font></span>)</span> <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="22_26"><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 <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22_26"/><i><font color="Green">E408</font></i>: 
<span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E2:22_26"/><i><font color="Green">E409</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font>, <font color="Maroon">fp</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E4:22_26"/>
not <font color="Maroon">p</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="abian.html#D5">ABIAN:def 5</a></i>;<br/>
<a NAME="E5:22_26"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="abian.html#D4">ABIAN:def 4</a></i>;<br/>
<a NAME="E6:22_26"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">fp</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="brouwer.html#K1" target="_self">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E7:22_26"/><b>then </b><i><font color="Green">E411</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">V1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E9:22_26"/><i><font color="Green">E412</font></i>: 
<font color="Maroon">V1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E10:22_26"/><i><font color="Green">E413</font></i>: 
<font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<a NAME="E11:22_26"/><i><font color="Green">E414</font></i>: 
<span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:22_26"/>
<a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="tops_2.html#T70">TOPS_2:70</a>, <a class="ref" href="topreala.html#T56">TOPREALA:56</a></i>;<br/>
<a NAME="E13:22_26"/><b>then </b><i><font color="Green">E415</font></i>: 
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="topgrp_1.html#T25">TOPGRP_1:25</a></i>;<br/>
<a NAME="E14:22_26"/><i><font color="Green">E416</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:22_26"/><i><font color="Green">E417</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topalg_2.html#K3">R^1</a> ,<a href="topalg_2.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:22_26"/><b>then </b><i><font color="Green">E418</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> 
 
;<br/>
<a NAME="E17:22_26"/><b>then </b><i><font color="Green">E419</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E18:22_26"/><i><font color="Green">E420</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <a href="topreala.html#K2">R2Homeomorphism</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp;  <a href="topreala.html#K2">R2Homeomorphism</a>  is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a>, <a class="ref" href="topreala.html#T56">TOPREALA:56</a></i>;<br/>
<i><font color="Green">E421</font></i>: <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font>
<b>by </b><i><a class="txt" href="brouwer.html#E6:22"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
;<br/>
<a NAME="E20:22_26"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<a NAME="E21:22_26"/><b>then </b><i><font color="Green">E422</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E22:22_26"/>
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="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_26_2"/><i><font color="Green">E423</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:22_26_2"/><i><font color="Green">E424</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="brouwer.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E23:22_26"/><b>then </b><i><font color="Green">E425</font></i>: 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E8:22"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E24:22_26"/>
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">fp</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 
<b>by </b><i>, , <a class="ref" href="brouwer.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E25:22_26"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">fp</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E26:22_26"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">fp</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, <a class="txt" href="brouwer.html#E4:22"><i><font color="Green">E62</font></i></a>, <a class="ref" href="brouwer.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span><b> such that </b><br/><a NAME="E28:22_26"/><i><font color="Green">E426</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">fp</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E29:22_26"/><i><font color="Green">E427</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E30:22_26"/><i><font color="Green">E428</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font>
 <b>by </b><i><a class="txt" href="brouwer.html#E2:22"><i><font color="Green">E54</font></i></a>, <a class="txt" href="brouwer.html#E3:22"><i><font color="Green">E58</font></i></a>, <a class="txt" href="brouwer.html#E7:22"><i><font color="Green">E64</font></i></a>, , , <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>consider </b><font color="Maroon">WW</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E32:22_26"/><i><font color="Green">E430</font></i>: 
<font color="Maroon">WW</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E33:22_26"/><i><font color="Green">E431</font></i>: 
<font color="Maroon">WW</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="brouwer.html#E11:22"><i><font color="Green">E68</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>consider </b><font color="Maroon">SF</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E35:22_26"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">WW</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">SF</font>
 <b>and </b><br/><a NAME="E36:22_26"/><i><font color="Green">E434</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> holds <br/> ex <font color="Olive">X1</font>, <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<a NAME="E37:22_26"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">fp</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">WW</font>
 
<b>by </b><i><a class="ref" href="brouwer.html#D3" target="_self">Def3</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<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="E39:22_26"/><i><font color="Green">E438</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">fp</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E40:22_26"/><i><font color="Green">E439</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">X1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">Y1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E42:22_26"/><i><font color="Green">E440</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E43:22_26"/><i><font color="Green">E441</font></i>: 
<font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E44:22_26"/><i><font color="Green">E442</font></i>: 
<font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">ZZ</font> = <font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">XX</font> = <font color="Maroon">X1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>, <font color="Maroon">YY</font> = <font color="Maroon">Y1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span> <b>by </b><i>, <a class="txt" href="brouwer.html#E16:22"><i><font color="Green">E78</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<a NAME="E46:22_26"/>
<font color="Maroon">fp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 
<b>by </b><i>, <a class="txt" href="brouwer.html#E15:22"><i><font color="Green">E76</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E47:22_26"/><b>then </b>
<font color="Maroon">fp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YY</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">M</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span><b> such that </b><br/><a NAME="E49:22_26"/><i><font color="Green">E446</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E50:22_26"/><i><font color="Green">E447</font></i>: 
<font color="Maroon">M</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E51:22_26"/><i><font color="Green">E448</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">YY</font>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>take </b><font color="Maroon">W1</font> = <font color="Maroon">XX</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">M</font>;<br/>

<a NAME="E52:22_26"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, <a class="txt" href="brouwer.html#E15:22"><i><font color="Green">E76</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E53:22_26"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E54:22_26"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>thus </b><a NAME="E55:22_26"/>
<font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="brouwer.html#E18:22"><i><font color="Green">E80</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>

<b>let </b><font color="Maroon">b</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="E56:22_26"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span><b> such that </b><br/><a NAME="E58:22_26"/><i><font color="Green">E449</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E59:22_26"/><i><font color="Green">E450</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">fa</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E61:22_26"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E62:22_26"/><b>then </b><i><font color="Green">E452</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E63:22_26"/>
not <font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="abian.html#D5">ABIAN:def 5</a></i>;<br/>
<a NAME="E64:22_26"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="abian.html#D4">ABIAN:def 4</a></i>;<br/>
<a NAME="E65:22_26"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">fa</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="brouwer.html#K1" target="_self">DiffElems</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E66:22_26"/><b>then </b><i><font color="Green">E453</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">OK</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E67:22_26"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E68:22_26"/><b>then </b>
<font color="Maroon">fa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E69:22_26"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E70:22_26"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">fa</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E15:22"><i><font color="Green">E76</font></i></a>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E71:22_26"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">fa</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E72:22_26"/><b>then </b><i><font color="Green">E454</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">fa</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E73:22_26"/><i><font color="Green">E455</font></i>: 
<a href="topreala.html#K2">R2Homeomorphism</a>  <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> 
 
<b>by </b><i><a class="txt" href="brouwer.html#E6:22"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E74:22_26"/><i><font color="Green">E456</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E6:22"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tops_2.html#T62">TOPS_2:62</a></i>;<br/>
<a NAME="E75:22_26"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">WW</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E76:22_26"/><b>then </b>
<font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">WW</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E77:22_26"/><b>then </b>
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">OK</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="brouwer.html#D3" target="_self">Def3</a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E78:22_26"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">fa</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<a NAME="E79:22_26"/><b>then </b>
<a href="topreala.html#K2">R2Homeomorphism</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">fa</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="hidden.html#R2">in</a> <a href="topreala.html#K2">R2Homeomorphism</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="brouwer.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E80:22_26"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">fa</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <a href="topreala.html#K2">R2Homeomorphism</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E81:22_26"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">fa</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E6:22"><i><font color="Green">E63</font></i></a>, , , <a class="ref" href="partfun3.html#T1">PARTFUN3:1</a></i>;<br/>
<a NAME="E82:22_26"/><b>then </b>
<span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E83:22_26"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E80:22"/>
 <a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>


</div>
