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<b>let </b><font color="Maroon" title="c1">X</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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:137_1_1"/><i><font color="Green" title="E83">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E2:137_1_1"/><b>then </b><i><font color="Green" title="E84">A2</font></i>: 
( <font color="Maroon" title="c2">x</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#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c2">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="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">x</font> as   <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> <b>by </b><i><a class="txt" href="pscomp_1.html#E1:137_1_1"><i><font color="Green" title="E83">A1</font></i></a></i>;<br/><a NAME="E4:137_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T95" target="_self" title="PSCOMP_1:th.95">Th95</a></i>;<br/><a NAME="E5:137_1_1"/><b>then </b><i><font color="Green" title="E85">A3</font></i>: 
<font color="Maroon" title="c3">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#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:137_1_1"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></i>;<br/><a NAME="E6:137_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-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#T97" target="_self" title="PSCOMP_1:th.97">Th97</a></i>;<br/><a NAME="E7:137_1_1"/><b>then </b><i><font color="Green" title="E86">A4</font></i>: 
<font color="Maroon" title="c3">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">N-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="pscomp_1.html#E2:137_1_1"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></i>;<br/><a NAME="E8:137_1_1"/><i><font color="Green" title="E87">A5</font></i>: 
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</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#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E9:137_1_1"/>
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-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">N-most</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T101" target="_self" title="PSCOMP_1:th.101">Th101</a></i>;<br/><a NAME="E10:137_1_1"/><b>then </b>
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</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#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:137_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:137_1_1"><i><font color="Green" title="E87">A5</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a></i>;<br/><a NAME="E12:137_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">N-most</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:137_1_1"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E13:137_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</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="c3">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T98" target="_self" title="PSCOMP_1:th.98">Th98</a></i>;<br/><a NAME="E14:137_1_1"/><b>then </b><i><font color="Green" title="E88">A6</font></i>: 
<font color="Maroon" title="c3">p</font> <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#K29" title="PSCOMP_1:func.29">N-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="pscomp_1.html#E7:137_1_1"><i><font color="Green" title="E86">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E15:137_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E16:137_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:137_1_1"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="pscomp_1.html#E14:137_1_1"><i><font color="Green" title="E88">A6</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:137_1"/><i><font color="Green" title="E83">A7</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E3:137_1"/>
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-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">N-most</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T101" target="_self" title="PSCOMP_1:th.101">Th101</a></i>;<br/><a NAME="E4:137_1"/><b>then </b><i><font color="Green" title="E84">A8</font></i>: 
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-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="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:137_1"/>
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</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#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><b>hence </b><a NAME="E6:137_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:137_1"><i><font color="Green" title="E83">A7</font></i></a>, <a class="txt" href="pscomp_1.html#E4:137_1"><i><font color="Green" title="E84">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:137"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">N-min</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:137_2_1"/><i><font color="Green" title="E83">A9</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E2:137_2_1"/><b>then </b><i><font color="Green" title="E84">A10</font></i>: 
( <font color="Maroon" title="c2">x</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#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c2">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="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">x</font> as   <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> <b>by </b><i><a class="txt" href="pscomp_1.html#E1:137_2_1"><i><font color="Green" title="E83">A9</font></i></a></i>;<br/><a NAME="E4:137_2_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T95" target="_self" title="PSCOMP_1:th.95">Th95</a></i>;<br/><a NAME="E5:137_2_1"/><b>then </b><i><font color="Green" title="E85">A11</font></i>: 
<font color="Maroon" title="c3">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">N-max</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:137_2_1"><i><font color="Green" title="E84">A10</font></i></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></i>;<br/><a NAME="E6:137_2_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</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#T97" target="_self" title="PSCOMP_1:th.97">Th97</a></i>;<br/><a NAME="E7:137_2_1"/><b>then </b><i><font color="Green" title="E86">A12</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</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="c3">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:137_2_1"><i><font color="Green" title="E84">A10</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></i>;<br/><a NAME="E8:137_2_1"/><i><font color="Green" title="E87">A13</font></i>: 
 <a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</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#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E9:137_2_1"/>
 <a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</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">N-most</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T101" target="_self" title="PSCOMP_1:th.101">Th101</a></i>;<br/><a NAME="E10:137_2_1"/><b>then </b>
 <a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</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#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:137_2_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">NW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:137_2_1"><i><font color="Green" title="E87">A13</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a></i>;<br/><a NAME="E12:137_2_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">N-most</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:137_2_1"><i><font color="Green" title="E84">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E13:137_2_1"/><b>then </b>
<font color="Maroon" title="c3">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#K30" title="PSCOMP_1:func.30">N-max</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#T98" target="_self" title="PSCOMP_1:th.98">Th98</a></i>;<br/><a NAME="E14:137_2_1"/><b>then </b><i><font color="Green" title="E88">A14</font></i>: 
<font color="Maroon" title="c3">p</font> <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#K30" title="PSCOMP_1:func.30">N-max</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="pscomp_1.html#E7:137_2_1"><i><font color="Green" title="E86">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E15:137_2_1"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E16:137_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:137_2_1"><i><font color="Green" title="E85">A11</font></i></a>, <a class="txt" href="pscomp_1.html#E14:137_2_1"><i><font color="Green" title="E88">A14</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:137_2"/><i><font color="Green" title="E83">A15</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E3:137_2"/>
 <a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</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">N-most</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T101" target="_self" title="PSCOMP_1:th.101">Th101</a></i>;<br/><a NAME="E4:137_2"/><b>then </b><i><font color="Green" title="E84">A16</font></i>: 
 <a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</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="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:137_2"/>
 <a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</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#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><b>hence </b><a NAME="E6:137_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:137_2"><i><font color="Green" title="E83">A15</font></i></a>, <a class="txt" href="pscomp_1.html#E4:137_2"><i><font color="Green" title="E84">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:137"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">NE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30" title="PSCOMP_1:func.30">N-max</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div>
