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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</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><a NAME="E1:151"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K26" target="_self">S-most</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:151"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K19" target="_self">SW-corner</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" target="_self">SE-corner</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E3:151"/><b>then </b><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19" target="_self">SW-corner</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" target="_self">SE-corner</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:151"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K19" target="_self">SW-corner</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" target="_self">S-bound</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</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" target="_self">S-bound</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" target="_self">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" target="_self">S-bound</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" target="_self">SE-corner</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" target="_self">S-bound</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T73" target="_self">Th73</a>, <a class="ref" href="pscomp_1.html#T79" target="_self">Th79</a>, <a class="ref" href="pscomp_1.html#T114" target="_self">Th114</a></i>;<br/>
<b>hence </b><a NAME="E5:151"/>
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</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="pscomp_1.html#E3:151"><i><font color="Green">E114</font></i></a>, <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>;<br/>

<b>assume </b><a NAME="E6:151"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6">compact</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  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="E8:151"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K10" target="_self">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K26" target="_self">S-most</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" target="_self">S-max</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K11" target="_self">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K26" target="_self">S-most</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T116" target="_self">Th116</a></i>;<br/>
<b>hence </b><a NAME="E9:151"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <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" target="_self">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="pscomp_1.html#E1:151"><i><font color="Green">E113</font></i></a>, <a class="txt" href="pscomp_1.html#E108"><i><font color="Green">Lemma75</font></i></a></i>;<br/>


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