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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</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>, <font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</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="E1:34"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T19">COMPTS_1:19</a></i>;<br/>
<b>assume </b><a NAME="E2:34"/><i><font color="Green">E39</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">X</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E3:34"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:34"/><i><font color="Green">E41</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:34"/>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T15">SPRECT_1:15</a></i>;<br/>
<a NAME="E7:34"/><b>then </b>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K23">W-most</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="sprect_2.html#T16">SPRECT_2:16</a></i>;<br/>
<a NAME="E8:34"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span></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="pscomp_1.html#K27">W-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T88">PSCOMP_1:88</a></i>;<br/>
<a NAME="E9:34"/><i><font color="Green">E44</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="sprect_1.html#T15">SPRECT_1:15</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
(  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:34_1_1"/><b>then </b>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K23">W-most</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="sprect_2.html#T16">SPRECT_2:16</a></i>;<br/><a NAME="E3:34_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span></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="pscomp_1.html#K27">W-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T88">PSCOMP_1:88</a></i>;<br/><a NAME="E4:34_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E5:34_1_1"/>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_2"/>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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