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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">S-most</a> <font color="Olive" title="b1">Z</font> holds <br/>( <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Olive" title="b1">Z</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; ( <font color="Olive" title="b1">Z</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> implies ( <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Olive" title="b1">Z</font></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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K29" title="PSCOMP_1:func.29">S-max</a> <font color="Olive" title="b1">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">Z</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">S-most</a> <font color="Maroon" title="c2">Z</font> implies ( <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; ( <font color="Maroon" title="c2">Z</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> implies ( <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Maroon" title="c2">Z</font></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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K29" title="PSCOMP_1:func.29">S-max</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) ) ) )</span><br/>



<b>assume </b><a NAME="E1:145"/><i><font color="Green" title="E88">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">S-most</a> <font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; ( <font color="Maroon" title="c2">Z</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> implies ( <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Maroon" title="c2">Z</font></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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K29" title="PSCOMP_1:func.29">S-max</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) ) )</span><br/>

<a NAME="E2:145"/><b>then </b><i><font color="Green" title="E89">A2</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">SW-corner</a> <font color="Maroon" title="c2">Z</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">SE-corner</a> <font color="Maroon" title="c2">Z</font></span>)</span> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:145"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">SW-corner</a> <font color="Maroon" title="c2">Z</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#K14" title="PSCOMP_1:func.14">S-bound</a> <font color="Maroon" title="c2">Z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Maroon" title="c2">Z</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#K14" title="PSCOMP_1:func.14">S-bound</a> <font color="Maroon" title="c2">Z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">S-max</a> <font color="Maroon" title="c2">Z</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#K14" title="PSCOMP_1:func.14">S-bound</a> <font color="Maroon" title="c2">Z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">SE-corner</a> <font color="Maroon" title="c2">Z</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#K14" title="PSCOMP_1:func.14">S-bound</a> <font color="Maroon" title="c2">Z</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<b>hence </b><a NAME="E4:145"/>
<font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="#E2:145"><i><font color="Green" title="E89">A2</font></i></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">Z</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> implies ( <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Maroon" title="c2">Z</font></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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K29" title="PSCOMP_1:func.29">S-max</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) )</span><br/>

<b>assume </b><a NAME="E5:145"/>
<font color="Maroon" title="c2">Z</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Maroon" title="c2">Z</font></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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K29" title="PSCOMP_1:func.29">S-max</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">Z</font> = <font color="Maroon" title="c2">Z</font> as  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> ;<br/>
<a NAME="E7:145"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">proj1</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">S-most</a> <font color="Maroon" title="c3">Z</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">S-max</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">proj1</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">S-most</a> <font color="Maroon" title="c3">Z</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<b>hence </b><a NAME="E8:145"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">S-min</a> <font color="Maroon" title="c2">Z</font></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> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K29" title="PSCOMP_1:func.29">S-max</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="#E1:145"><i><font color="Green" title="E88">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E112"><i><font color="Green" title="E69">Lm9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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