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<b>let </b><font color="Maroon">P1</font> be   <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>, <font color="Maroon">P2</font> be   <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>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">f2</font> be   <a href="struct_0.html#NM7">FinSequence</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">p1</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>, <font color="Maroon">p2</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>set </b><font color="Maroon">P</font> =  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>
<b>assume </b><a NAME="E1:72"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font> )
 ;<br/>

<a NAME="E2:72"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E3:72"/><b>then </b><i><font color="Green">E30</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E4:72"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:72"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:72"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E7:72"/><b>then </b><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E8:72"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:72"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:72"/>
( <font color="Maroon">P1</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">P2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; not <font color="Maroon">P1</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">P2</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i>, , , <a class="ref" href="jgraph_6.html#T14" target="_self">Th14</a>, , , , <a class="ref" href="topreal1.html#T18">TOPREAL1:18</a></i>;<br/>


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