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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="topreal1.html#V1">special</a> <a href="struct_0.html#NM8">FinSequence</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="E1:42"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="jordan12.html#R2" target="_self">are_in_general_position</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:42"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E3:42"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:42"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="jordan12.html#R1" target="_self">is_in_general_position_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:42"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan12.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:42"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="jordan12.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:42"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="jordan12.html#R1" target="_self">is_in_general_position_wrt</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:42"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan12.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:42"/><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:42"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E3:42"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:42"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E7:42"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E10:42"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E8:42"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E11:42"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E8:42"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E12:42"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <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> <b>by </b><i><a class="txt" href="jordan12.html#E11:42"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E14:42"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E9:42"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <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> <b>by </b><i><a class="txt" href="jordan12.html#E12:42"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E16:42"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E9:42"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E17:42"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E10:42"><i><font color="Green">E50</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E18:42"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is  <a href="connsp_1.html#V2">connected</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>
 
<b>by </b><i><a class="ref" href="goboard9.html#T8">GOBOARD9:8</a></i>;<br/>
<a NAME="E19:42"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E10:42"><i><font color="Green">E50</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<a NAME="E20:42"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E16:42"><i><font color="Green">E53</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<a NAME="E21:42"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:42"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan12.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E22:42"/><b>then </b><i><font color="Green">E58</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E23:42"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:42"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan12.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E24:42"/><b>then </b><i><font color="Green">E60</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E25:42"/><i><font color="Green">E61</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E26:42"/>
(  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="sppol_1.html#V2">vertical</a> or  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="sppol_1.html#V1">horizontal</a> )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T41">SPPOL_1:41</a></i>;<br/>
<a NAME="E27:42"/><b>then </b><i><font color="Green">E62</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jordan12.html#E25:42"><i><font color="Green">E61</font></i></a>, <a class="ref" href="sppol_1.html#T36">SPPOL_1:36</a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1"/>
(  not <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/><i><font color="Green">E63</font></i>: 
not <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:42_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E3:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan12.html#E4:42"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self">Def1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:42_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:42_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><a NAME="E5:42_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E4:42_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <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/><a NAME="E7:42_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan12.html#E8:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan12.html#E9:42"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E8:42_1_1"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></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">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan12.html#E4:42_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E9:42_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E8:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan12.html#E9:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan12.html#E7:42_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E1:42_1_1_1"><i><font color="Green">E69</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan12.html#E5:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan12.html#E9:42_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan12.html#E5:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan12.html#E9:42_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E11:42_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan12.html#E6:42"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E12:42_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/><a NAME="E13:42_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E25:42"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan12.html#E11:42_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><b>hence </b><a NAME="E14:42_1_1"/>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> iff ex <font color="Olive">b<sub>1</sub></font> being  <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E27:42"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan12.html#E4:42_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan12.html#E7:42_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan12.html#E8:42_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E10:42_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan12.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan12.html#T33" target="_self">Th33</a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:42_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/><a NAME="E3:42_1_2"/><i><font color="Green">E65</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:42_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:42_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
(  not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E18:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E19:42"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E20:42"><i><font color="Green">E56</font></i></a>, <a class="ref" href="jordan1j.html#T36">JORDAN1J:36</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_2_1_1"/>
(  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E18:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan12.html#E19:42"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan12.html#E20:42"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan12.html#E3:42_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan1j.html#T36">JORDAN1J:36</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_2_1_1_1"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:42_1_2_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan12.html#E22:42"><i><font color="Green">E58</font></i></a>, <a class="ref" href="gobrd14.html#T27">GOBRD14:27</a></i>;<br/><a NAME="E3:42_1_2_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan12.html#E22:42"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan12.html#E23:42"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan12.html#E24:42"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E4:42_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E1:42_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="gobrd14.html#T27">GOBRD14:27</a></i>;<br/><a NAME="E4:42_1_2_1_1_1"/>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/><b>hence </b><a NAME="E5:42_1_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E2:42_1_2_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan12.html#E3:42_1_2_1_1_1"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_2_1_1_2"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:42_1_2_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan12.html#E23:42"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan12.html#E24:42"><i><font color="Green">E60</font></i></a>, <a class="ref" href="gobrd14.html#T28">GOBRD14:28</a></i>;<br/><a NAME="E3:42_1_2_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan12.html#E21:42"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan12.html#E22:42"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan12.html#E23:42"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan12.html#E24:42"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan12.html#E4:42_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan12.html#E1:42_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="gobrd14.html#T28">GOBRD14:28</a></i>;<br/><a NAME="E4:42_1_2_1_1_2"/>
 <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D2">GOBOARD9:def 2</a></i>;<br/><b>hence </b><a NAME="E5:42_1_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E2:42_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan12.html#E3:42_1_2_1_1_2"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:42_1_2"/>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> iff ex <font color="Olive">b<sub>1</sub></font> being  <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan12.html#E2:42_1_2"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
