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

<b>reconsider </b><font color="Maroon">fX2</font> =  <a href="jordan.html#K7" target="_self">Proj2_1</a>  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="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E2:93"/>
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="topmetr.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="93_1"><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="topmetr.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:93_1"/><i><font color="Green">E69</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="E2:93_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E3:93_1"/><i><font color="Green">E74</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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K1" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E4:93_1"/><i><font color="Green">E78</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> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E2"><i><font color="Green">Lemma62</font></i></a></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="txt" href="jordan.html#E2:93_1"><i><font color="Green">E70</font></i></a>, <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="E7:93_1"/><i><font color="Green">E79</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E8:93_1"/><i><font color="Green">E80</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">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="txt" href="jordan.html#E1:93_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan.html#T3" target="_self">Th3</a>, <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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="E10:93_1"/>
<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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="93_1_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="E1:93_1_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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="E2:93_1_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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="E3:93_1_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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="E12:93_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <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">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="E13:93_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <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"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="jordan.html#T4" target="_self">Th4</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:93_1"/><b>then </b>
<font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E15:93_1"/>
<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="jordan.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<a NAME="E16:93_1"/>
<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="E17:93_1"/>
<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="E18:93_1"/>
<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="E20:93_1"/><i><font color="Green">E82</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="E21:93_1"/><i><font color="Green">E83</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/>
<a NAME="E22:93_1"/><i><font color="Green">E109</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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="jordan.html#K1" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E23:93_1"/><i><font color="Green">E110</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> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E2"><i><font color="Green">Lemma62</font></i></a></i>;<br/>
<a NAME="E24:93_1"/>
<font color="Maroon">a</font> <a href="jordan.html#K2" target="_self">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E4:93_1"><i><font color="Green">E78</font></i></a>, <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="E26:93_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">a</font> <a href="jordan.html#K2" target="_self">`2</a>  <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="E27:93_1"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&lt;</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E28:93_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font>
 ;<br/>
<a NAME="E29:93_1"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:93_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E30:93_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="txt" href="jordan.html#E12:93_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan.html#E20:93_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E31:93_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="E32:93_1"/><b>then </b>
 <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">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&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="E33:93_1"/><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">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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#R1">&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="E34:93_1"/><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"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
;<br/>
<a NAME="E35:93_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <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="txt" href="jordan.html#E21:93_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<b>hence </b><a NAME="E36:93_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="jordan.html#E3:93_1"><i><font color="Green">E74</font></i></a>, , <a class="txt" href="jordan.html#E7:93_1"><i><font color="Green">E79</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:93"/>
 <a href="jordan.html#K7" target="_self">Proj2_1</a>  is  <a href="pre_topc.html#V5">continuous</a> <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="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>


</div>
