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<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>assume </b><a NAME="E1:78"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="jgraph_6.html#K1">inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="jgraph_6.html#K3">outside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<a NAME="E2:78"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>8</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>9</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>10</sub></font> is   <a href="real_1.html#NM1">Real</a> )
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:78"/><i><font color="Green">E57</font></i>: 
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan.html#E1:78"><i><font color="Green">E55</font></i></a>, <a class="ref" href="sppol_2.html#T58">SPPOL_2:58</a></i>;<br/>
<a NAME="E4:78"/><i><font color="Green">E58</font></i>: 
 <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="jgraph_6.html#K3">outside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="jordan.html#T48" target="_self">Th48</a></i>;<br/>
<b>thus </b> <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jgraph_6.html#K3">outside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="jordan.html#T47" target="_self">Th47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_6.html#K3">outside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="txt" href="jordan.html#E1:78"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan.html#E2:78"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan.html#E3:78"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan1.html#T49">JORDAN1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_6.html#K3">outside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T47" target="_self">Th47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_6.html#K1">inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#K3">outside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan.html#E1:78"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan.html#E2:78"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan.html#E3:78"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan1.html#T41">JORDAN1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#K1">inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#K3">outside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#K1">inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="jordan.html#E4:78"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="jgraph_6.html#K1">inside_of_rectangle</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="jordan.html#T49" target="_self">Th49</a></i>
;<br/>


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