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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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/>

<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green">E14</font></i>: 
 <a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:21"/><i><font color="Green">E15</font></i>: 
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:21"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pscomp_1.html#T131">PSCOMP_1:131</a></i>;<br/>
<a NAME="E4:21"/><i><font color="Green">E16</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E5:21"/>
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1j.html#E2:21"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E6:21"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E3:21"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan1j.html#E4:21"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E7:21"/><b>then </b>
 <a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K26">S-most</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E2:21"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan1j.html#E4:21"><i><font color="Green">E16</font></i></a>, <a class="ref" href="sprect_2.html#T15">SPRECT_2:15</a></i>;<br/>
<a NAME="E8:21"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T118">PSCOMP_1:118</a></i>;<br/>
<a NAME="E9:21"/>
 <a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T14">SPRECT_1:14</a></i>;<br/>
<a NAME="E10:21"/><b>then </b>
 <a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K26">S-most</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:21"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan1j.html#E4:21"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan1j.html#E6:21"><i><font color="Green">E17</font></i></a>, <a class="ref" href="sprect_2.html#T15">SPRECT_2:15</a></i>;<br/>
<a NAME="E11:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T118">PSCOMP_1:118</a></i>;<br/>
<a NAME="E12:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1j.html#E8:21"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E13:21"/>
 <a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K33">S-max</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1j.html#E4:21"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan1j.html#E6:21"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>


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