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<span class="kw">let </span><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#V2" title="COMPTS_1:attr.2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">and </span><br/><a NAME="E2:20"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font></span><br/>

<a NAME="E3:20"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E4:20"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E5:20"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E6:20"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="pscomp_1.html#T131" title="PSCOMP_1:th.131">PSCOMP_1:131</a></span>;<br/>
<a NAME="E7:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A7</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E8:20"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">S-most</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E9:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`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#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T118" title="PSCOMP_1:th.118">PSCOMP_1:118</a></span>;<br/>
<a NAME="E10:20"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T14" title="SPRECT_1:th.14">SPRECT_1:14</a></span>;<br/>
<a NAME="E11:20"/><span class="kw">then </span>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">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#K17" title="PSCOMP_1:func.17">S-most</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E12:20"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`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#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T118" title="PSCOMP_1:th.118">PSCOMP_1:118</a></span>;<br/>
<a NAME="E13:20"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:20"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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