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

<b>let </b><font color="Maroon">K0</font> be   <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">B0</font> be   <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>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B0</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:42"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K0</font> &amp; <font color="Maroon">B0</font> <a href="hidden.html#R1">=</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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">K0</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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> : ( ( ( <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  )
 ;<br/>

<a NAME="E2:42"/>
( ( ( <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  ) or ( <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp;  <a href="jordan2c.html#K5">1.REAL</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 
<b>by </b><i>, <a class="ref" href="revrot_1.html#T19">REVROT_1:19</a></i>;<br/>
<a NAME="E3:42"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="jordan2c.html#K5">1.REAL</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E5:42"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K1</font> = <font color="Maroon">K0</font> as  non <a href="xboole_0.html#V1">empty</a> <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="E5:42"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">K0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p8</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="E3:42_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p8</font> &amp; ( ( <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:42"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:42_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E7:42"><i><font color="Green">E45</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:42_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B0</font>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jgraph_2.html#E7:42"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:42"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/>
<a NAME="E7:42"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:42_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
 ;<br/>

<a NAME="E2:42_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:42_2"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E2:42_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:42_2"/><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="hidden.html#R1">=</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:42_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> 
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E3:42_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:42_2"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <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> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> 
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:42_2"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <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">(<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#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:42_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E2:42_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jgraph_2.html#E4:42_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jgraph_2.html#E8:42"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:42"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">K1</font> <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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:42_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p8</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="E3:42_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">p8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; ( ( <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:42"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:42_3"/>
( <font color="Maroon">z</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> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:42_3"/>
<font color="Maroon">z</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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:42"/><i><font color="Green">E53</font></i>: 
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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<i><font color="Green">E54</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_2.html#E7:42"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E11:42"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g2</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E13:42"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/>
<a NAME="E14:42"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:42_5"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
 ;<br/>

<a NAME="E2:42_5"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:42_5"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:42_5"/><i><font color="Green">E60</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K13">proj1</a>  <a href="hidden.html#R1">=</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:42_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:42_5"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <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> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> 
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:42_5"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <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">(<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#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:42_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jgraph_2.html#E9:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jgraph_2.html#E10:42"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E65</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_2.html#E8:42"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E16:42"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g1</font> = <a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E18:42"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">q</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 <font color="Olive">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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>assume </b><a NAME="E1:42_7"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
 ;<br/>

the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>then consider </b><font color="Maroon">p3</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="E4:42_7"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; ( ( <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jgraph_2.html#E9:42"><i><font color="Green">E53</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_7_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:42_7_2"/><b>then </b>
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_2.html#E10:42"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_7_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_2.html#E10:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_2.html#E13:42"><i><font color="Green">E55</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:42_7"/>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:42"/>
for <font color="Olive">p</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 <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> holds <br/><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_8"><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="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:42_8"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
 ;<br/>

<a NAME="E2:42_8"/><i><font color="Green">E149</font></i>: 
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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<i><font color="Green">E150</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="txt" href="jgraph_2.html#E10:42"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<i><font color="Green">E151</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>then consider </b><font color="Maroon">p3</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="E6:42_8"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; ( ( <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jgraph_2.html#E9:42"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E7:42_8"/><i><font color="Green">E153</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">p</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">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></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="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:42_8"/>
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E9:42"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>then </b><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> = 
<a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></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="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_2.html#E9:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jgraph_2.html#E13:42"><i><font color="Green">E55</font></i></a>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></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="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<b>hence </b><a NAME="E10:42_8"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">f2</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E21:42"/><i><font color="Green">E154</font></i>: 
for <font color="Olive">p</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 <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/>
<a NAME="E22:42"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="txt" href="jgraph_2.html#E9:42"><i><font color="Green">E53</font></i></a>, </i>;<br/>
<a NAME="E23:42"/>
for <font color="Olive">p</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 <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> holds <br/><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_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">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:42_9"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
 ;<br/>

<a NAME="E2:42_9"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">K1</font> <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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:42_9_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p8</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="E3:42_9_1"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">p8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; ( ( <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:42"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:42_9_1"/>
( <font color="Maroon">z</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> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:42_9_1"/>
<font color="Maroon">z</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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:42_9"/><i><font color="Green">E159</font></i>: 
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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<i><font color="Green">E160</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K1</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<i><font color="Green">E161</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>then consider </b><font color="Maroon">p3</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="E7:42_9"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; ( ( <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jgraph_2.html#E13:42"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E8:42_9"/><i><font color="Green">E163</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">p</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">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></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="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:42_9"/>
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E13:42"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>then </b><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> = 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></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="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_2.html#E13:42"><i><font color="Green">E55</font></i></a>, , , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></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="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<b>hence </b><a NAME="E11:42_9"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E25:42"/><i><font color="Green">E164</font></i>: 
for <font color="Olive">p</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 <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/>
<a NAME="E26:42"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="txt" href="jgraph_2.html#E13:42"><i><font color="Green">E55</font></i></a>, </i>;<br/>
<a NAME="E27:42"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <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> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</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; <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</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> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">s</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="E1:42_10"/><i><font color="Green">E166</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> )
 ;<br/>

<b>set </b><font color="Maroon">p99</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span>;<br/>
<i><font color="Green">E167</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>consider </b><font color="Maroon">p3</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="E4:42_10"/><i><font color="Green">E168</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; ( ( <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:42"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<a NAME="E5:42_10"/><i><font color="Green">E169</font></i>: 
( ( ( ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) implies <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> ) &amp; ( ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> ) or <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <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"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:42_10"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E13:42"><i><font color="Green">E55</font></i></a>, , </i>;<br/>
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K0</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_2.html#E9:42"><i><font color="Green">E53</font></i></a>, , , </i>
;<br/>
<b>hence </b><a NAME="E8:42_10"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_2.html#E5:42"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E28:42"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="txt" href="jgraph_2.html#E6:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jgraph_2.html#E10:42"><i><font color="Green">E54</font></i></a>, , </i>;<br/>


</div>
