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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<a NAME="E1:32"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T19" title="COMPTS_1:th.19">COMPTS_1:19</a></i>;<br/>
<b>assume </b><a NAME="E2:32"/><i><font color="Green" title="E20">A2</font></i>: 
 <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c2">Y</font>
 ;<br/>

<a NAME="E3:32"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
 <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="jordan1j.html#T25" target="_self" title="JORDAN1J:th.25">Th25</a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green" title="E22">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:32"/>
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T14" title="SPRECT_1:th.14">SPRECT_1:14</a></i>;<br/>
<a NAME="E7:32"/><b>then </b>
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-most</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="jordan1j.html#E3:32"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="jordan1j.html#E4:32"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="jordan1j.html#E5:32"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></i>;<br/>
<a NAME="E8:32"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="pscomp_1.html#T118" title="PSCOMP_1:th.118">PSCOMP_1:118</a></i>;<br/>
<a NAME="E9:32"/><i><font color="Green" title="E25">A7</font></i>: 
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="sprect_1.html#T14" title="SPRECT_1:th.14">SPRECT_1:14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1"/>
(  <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or  <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 </span><b>by </b><i><a class="txt" href="jordan1j.html#E9:32"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>suppose </b></a><a NAME="E1:32_1_1"/>
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/><div class="add"><a NAME="E2:32_1_1"/><b>then </b>
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-most</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="jordan1j.html#E2:32"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="jordan1j.html#E4:32"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="jordan1j.html#T25" target="_self" title="JORDAN1J:th.25">Th25</a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></i>;<br/><a NAME="E3:32_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T118" title="PSCOMP_1:th.118">PSCOMP_1:118</a></i>;<br/><a NAME="E4:32_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan1j.html#E8:32"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E5:32_1_1"/>
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="jordan1j.html#E2:32"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="jordan1j.html#E4:32"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="jordan1j.html#T25" target="_self" title="JORDAN1J:th.25">Th25</a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>suppose </b></a><a NAME="E1:32_1_2"/>
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_1_2"/>
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="jordan1j.html#E2:32"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="jordan1j.html#E3:32"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="jordan1j.html#E4:32"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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