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

<b>let </b><font color="Maroon">sn</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">K1</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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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>means </b><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sn</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span>;<br/>
<b>assume </b><a NAME="E1:38"/>
<font color="Maroon">K1</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="Maroon">sn</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span> </span>}</span> 
 ;<br/>

<a NAME="E2:38"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">K1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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="Maroon">S<sub>1</sub></font>[<font color="Olive">p7</font>] </span>}</span> 
 
;<br/>
<a NAME="E3:38"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">K1</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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> :  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p7</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="jgraph_2.html#S2">JGRAPH_2:sch 2</a>();<br/></i>
<b>set </b><font color="Maroon">K10</font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <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></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="jgraph_4.html#T4" target="_self">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g0</font> = <span class="p1">(<span class="default">2 <a href="jgraph_4.html#K1" target="_self">NormF</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <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></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="jgraph_4.html#T6" target="_self">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon">g2</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> <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></span>)</span>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E7:38"/><i><font color="Green">E38</font></i>: 
( ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</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> <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></span>)</span><br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sn</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font> ) &amp; <font color="Maroon">g2</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T33">JGRAPH_2:33</a></i>;<br/>
<b>consider </b><font color="Maroon">g3</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> <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></span>)</span>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E9:38"/><i><font color="Green">E39</font></i>: 
( ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</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> <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></span>)</span><br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> &amp; <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">g3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">r2</font> ) &amp; <font color="Maroon">g3</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T31">JGRAPH_2:31</a></i>;<br/>
<a NAME="E10:38"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K15">TOP-REAL</a> 2
 
<b>by </b><i><a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>;<br/>
<a NAME="E11:38"/><i><font color="Green">E41</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> holds  <font color="Maroon">g3</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"><font color="Maroon">sn</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">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/>

<font color="Maroon">g0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default">2 <a href="jgraph_4.html#K1" target="_self">NormF</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="jgraph_4.html#T8" target="_self">Th8</a>, <a class="ref" href="jgraph_4.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i/>
;<br/>
<a NAME="E2:38_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sn</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="jgraph_4.html#T8" target="_self">Th8</a></i>;<br/>
<font color="Maroon">g1</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> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="jgraph_4.html#T8" target="_self">Th8</a>, <a class="ref" href="jgraph_4.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">p</font> <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/>
<b>hence </b><a NAME="E4:38_1"/>
<font color="Maroon">g3</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"><font color="Maroon">sn</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_4.html#T7" target="_self">Th7</a>, <a class="ref" href="jgraph_4.html#T8" target="_self">Th8</a>, </i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g3</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="jgraph_4.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E13:38"/><i><font color="Green">E43</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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> :  <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Olive">p7</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">K1</font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">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="E1:38_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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> :  <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Olive">p7</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p7</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="E3:38_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">p7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Maroon">p7</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/>
 <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Maroon">p7</font> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p7</font>
<b>by </b><i><a class="ref" href="jordan2c.html#D10">JORDAN2C:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p7</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E5:38_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sn</font> <a href="real_1.html#K4">*</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p7</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E6:38_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jgraph_4.html#E13:38"><i><font color="Green">E43</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:38"/>
<font color="Maroon">K1</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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> :  <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Olive">p7</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:38_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">p9</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="E3:38_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p9</font> &amp; <font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">sn</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p9</font></span><a href="toprns_1.html#K5">.|</a></span> )
 <b>by </b><i><a class="txt" href="jgraph_4.html#E13:38"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E4:38_3"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p9</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
 <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Maroon">p9</font> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p9</font>
<b>by </b><i><a class="ref" href="jordan2c.html#D10">JORDAN2C:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p9</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E6:38_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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> :  <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Olive">p7</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:38"/><b>then </b>
<font color="Maroon">K1</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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> :  <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Olive">p7</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E16:38"/><b>then </b>
<font color="Maroon">K1</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T7" target="_self">Th7</a>, <a class="ref" href="jgraph_4.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>hence </b><a NAME="E17:38"/>
<font color="Maroon">K1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


</div>
