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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="chain_1.html#M1" target="_self">Grating</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> or <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1"><b>suppose </b></a><a NAME="E1:92_1_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:92_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:92_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:92_1_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> ) holds <br/> <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:92_1_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:92_1_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E1:92_1_1_1_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> iff ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E2:92_1_1_1_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> iff ( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="chain_1.html#T51" target="_self">Th51</a>, <a class="ref" href="chain_1.html#T54" target="_self">Th54</a></i>;<br/></div><b>end;</b></div><a NAME="E4:92_1_1_1_1"/><i><font color="Green">E72</font></i>: 
not  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="chain_1.html#T52" target="_self">Th52</a></i>;<br/><a NAME="E5:92_1_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> 0 is <a href="abian.html#V1">even</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:92_1_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:92_1_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="chain_1.html#E7:92_1_1_1_1"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E9:92_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:92_1_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="chain_1.html#E7:92_1_1_1_1"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E10:92_1_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E12:92_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E14:92_1_1_1_1"/><i><font color="Green">E75</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) or ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="chain_1.html#T34" target="_self">Th34</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E16:92_1_1_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; ( ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) or ( not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:92_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="chain_1.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E17:92_1_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E10:92_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E19:92_1_1_1_1"/><i><font color="Green">E77</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) or ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E20:92_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:92_1_1_1_1_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> ) holds <br/>( <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) holds <br/>( <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E10:92_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="chain_1.html#E14:92_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="chain_1.html#T48" target="_self">Th48</a></i>;<br/>
<b>then </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="chain_1.html#E14:92_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
2

;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:92_1_1_1_1_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>17</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:92_1_1_1_1_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; not <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E4:92_1_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="chain_1.html#E4:92_1_1_1_1_2"><i><font color="Green">E79</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E7:92_1_1_1_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:92_1_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E5:92_1_1_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:92_1_1_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>,  <a href="real_1.html#NM1">Real</a>] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E10:92_1_1_1_1_2"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:92_1_1_1_1_2_2"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<b>hence </b><a NAME="E2:92_1_1_1_1_2_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>23</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E12:92_1_1_1_1_2"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E10:92_1_1_1_1_2"><i><font color="Green">E83</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>,  <a href="real_1.html#NM1">Real</a>] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E13:92_1_1_1_1_2"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:92_1_1_1_1_2_3"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<b>hence </b><a NAME="E2:92_1_1_1_1_2_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>24</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E15:92_1_1_1_1_2"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E13:92_1_1_1_1_2"><i><font color="Green">E85</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E17:92_1_1_1_1_2"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:92_1_1_1_1_2_4"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<a NAME="E2:92_1_1_1_1_2_4"/><b>then </b>
( ( <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> ) )
 
<b>by </b><i><a class="txt" href="chain_1.html#E12:92_1_1_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="chain_1.html#E15:92_1_1_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:92_1_1_1_1_2_4"/>
<font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E14:92_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:92_1_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E5:92_1_1_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<b>then </b><i><font color="Green">E88</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/>
<a NAME="E20:92_1_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) or ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_1_1_1_2_6_1"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_6_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2_6_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_1_2_6_1_1"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E12:92_1_1_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="chain_1.html#E15:92_1_1_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/><b>hence </b><a NAME="E3:92_1_1_1_1_2_6_1_1"/>
( ( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> ) or ( not <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E14:92_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="chain_1.html#E9:92_1_1_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="chain_1.html#E1:92_1_1_1_1_2_6_1_1"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_6_1_2"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2_6_1_2"/><i><font color="Green">E89</font></i>: 
not <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_1_2_6_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E12:92_1_1_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="chain_1.html#E15:92_1_1_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E3:92_1_1_1_1_2_6_1_2"/>
not <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E7:92_1_1_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="chain_1.html#E1:92_1_1_1_1_2_6_1_2"><i><font color="Green">E89</font></i></a></i>;<br/><b>hence </b><a NAME="E4:92_1_1_1_1_2_6_1_2"/>
( ( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> ) or ( not <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="chain_1.html#E1:92_1_1_1_1_2_6_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="chain_1.html#E2:92_1_1_1_1_2_6_1_2"><i><font color="Green">E90</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font> as   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="chain_1.html#E2:92_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="chain_1.html#E19:92_1_1_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="chain_1.html#T34" target="_self">Th34</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E22:92_1_1_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E23:92_1_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E5:92_1_1_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E24:92_1_1_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>,  <a href="real_1.html#NM1">Real</a>] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E25:92_1_1_1_1_2"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>29</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:92_1_1_1_1_2_7"/>
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<b>hence </b><a NAME="E2:92_1_1_1_1_2_7"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>29</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E27:92_1_1_1_1_2"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E25:92_1_1_1_1_2"><i><font color="Green">E91</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>,  <a href="real_1.html#NM1">Real</a>] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E28:92_1_1_1_1_2"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:92_1_1_1_1_2_8"/>
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<b>hence </b><a NAME="E2:92_1_1_1_1_2_8"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>30</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>30</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E30:92_1_1_1_1_2"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E28:92_1_1_1_1_2"><i><font color="Green">E93</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>29</sub></font>, <font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>30</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E32:92_1_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E5:92_1_1_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<b>then </b><i><font color="Green">E95</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/>
<a NAME="E34:92_1_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) or ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>33</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_1_1_1_2_10_1"/>
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_10_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2_10_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_1_2_10_1_1"/><b>then </b>
( <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E27:92_1_1_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="chain_1.html#E30:92_1_1_1_1_2"><i><font color="Green">E94</font></i></a></i>;<br/><b>hence </b><a NAME="E3:92_1_1_1_1_2_10_1_1"/>
( ( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> ) or ( not <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E14:92_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="chain_1.html#E24:92_1_1_1_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="chain_1.html#E1:92_1_1_1_1_2_10_1_1"><i><font color="Green">E96</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_10_1_2"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2_10_1_2"/><i><font color="Green">E96</font></i>: 
not <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_1_2_10_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E27:92_1_1_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="chain_1.html#E30:92_1_1_1_1_2"><i><font color="Green">E94</font></i></a></i>;<br/><a NAME="E3:92_1_1_1_1_2_10_1_2"/>
not <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E22:92_1_1_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="chain_1.html#E1:92_1_1_1_1_2_10_1_2"><i><font color="Green">E96</font></i></a></i>;<br/><b>hence </b><a NAME="E4:92_1_1_1_1_2_10_1_2"/>
( ( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> ) or ( not <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>33</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="chain_1.html#E1:92_1_1_1_1_2_10_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="chain_1.html#E2:92_1_1_1_1_2_10_1_2"><i><font color="Green">E97</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>33</sub></font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>32</sub></font> as   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="chain_1.html#E2:92_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="chain_1.html#E33:92_1_1_1_1_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="chain_1.html#T34" target="_self">Th34</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>27</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>33</sub></font>
;<br/>

<a NAME="E36:92_1_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>34</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_1_1_1_2_11_1"/>
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_11_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2_11_1_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_1_2_11_1_1"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E12:92_1_1_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="chain_1.html#E15:92_1_1_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/><b>hence </b><a NAME="E3:92_1_1_1_1_2_11_1_1"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E10:92_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_11_1_2"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2_11_1_2"/>
not <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_1_2_11_1_2"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E12:92_1_1_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="chain_1.html#E15:92_1_1_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/><b>hence </b><a NAME="E3:92_1_1_1_1_2_11_1_2"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E10:92_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E37:92_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E38:92_1_1_1_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:92_1_1_1_1"><i><font color="Green">E71</font></i></a></i>;<br/>

<a NAME="E39:92_1_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>34</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_1_1_1_2_12_1"/>
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_12_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2_12_1_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_1_2_12_1_1"/><b>then </b>
( <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E27:92_1_1_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="chain_1.html#E30:92_1_1_1_1_2"><i><font color="Green">E94</font></i></a></i>;<br/><b>hence </b><a NAME="E3:92_1_1_1_1_2_12_1_1"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E10:92_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_12_1_2"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2_12_1_2"/>
not <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_1_2_12_1_2"/><b>then </b>
( <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E27:92_1_1_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="chain_1.html#E30:92_1_1_1_1_2"><i><font color="Green">E94</font></i></a></i>;<br/><b>hence </b><a NAME="E3:92_1_1_1_1_2_12_1_2"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E10:92_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E40:92_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E41:92_1_1_1_1_2"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:92_1_1_1_1"><i><font color="Green">E71</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_13"><b>now </b></a><div class="add"><a NAME="E1:92_1_1_1_1_2_13"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E2:92_1_1_1_1_2_13"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:92_1_1_1_1_2_13"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E12:92_1_1_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="chain_1.html#E15:92_1_1_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E4:92_1_1_1_1_2_13"/>
( not <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; not <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="chain_1.html#E4:92_1_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:92_1_1_1_1_2_13"/>
not <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:92_1_1_1_1_2_13"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E27:92_1_1_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="chain_1.html#E30:92_1_1_1_1_2"><i><font color="Green">E94</font></i></a></i>;<br/><b>thus </b><a NAME="E7:92_1_1_1_1_2_13"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E14:92_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="chain_1.html#E5:92_1_1_1_1_2"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E43:92_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>31</sub></font> or <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>32</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E44:92_1_1_1_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E17:92_1_1_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="chain_1.html#T32" target="_self">Th32</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>34</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E45:92_1_1_1_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>34</sub></font> as   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E47:92_1_1_1_1_2"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E3:92_1_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="chain_1.html#E45:92_1_1_1_1_2"><i><font color="Green">E96</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>36</sub></font>, <font color="Maroon">c<sub>37</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E49:92_1_1_1_1_2"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font> &amp; ( ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) or ( not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:92_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="chain_1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E50:92_1_1_1_1_2"/>
<font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E47:92_1_1_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="chain_1.html#E49:92_1_1_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>38</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E52:92_1_1_1_1_2"/><i><font color="Green">E99</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) or ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E49:92_1_1_1_1_2"><i><font color="Green">E98</font></i></a></i>;<br/>
<a NAME="E53:92_1_1_1_1_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>38</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font> ) holds <br/>( <font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 
<b>by </b><i><a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="chain_1.html#E47:92_1_1_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="chain_1.html#E49:92_1_1_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="chain_1.html#E52:92_1_1_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="chain_1.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E54:92_1_1_1_1_2"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>38</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> ) holds <br/>( <font color="Maroon">c<sub>36</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E14:92_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="chain_1.html#E47:92_1_1_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="chain_1.html#E49:92_1_1_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="chain_1.html#E52:92_1_1_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="chain_1.html#T48" target="_self">Th48</a></i>;<br/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="chain_1.html#E52:92_1_1_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="chain_1.html#E53:92_1_1_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
1

;<br/>
<b>then consider </b><font color="Maroon">c<sub>39</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E57:92_1_1_1_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>39</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<a NAME="E58:92_1_1_1_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E57:92_1_1_1_1_2"><i><font color="Green">E102</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>40</sub></font> = <font color="Maroon">c<sub>39</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E60:92_1_1_1_1_2"/>
( <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font> &amp; not <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E58:92_1_1_1_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E61:92_1_1_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E54:92_1_1_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<i><font color="Green">E105</font></i>: <font color="Maroon">c<sub>38</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>38</sub></font>
<b>by </b><i><a class="txt" href="chain_1.html#E53:92_1_1_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>40</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="chain_1.html#E57:92_1_1_1_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_16"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_1_1_1_2_16"/>
( <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E4:92_1_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="chain_1.html#E61:92_1_1_1_1_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="chain_1.html#E62:92_1_1_1_1_2"><i><font color="Green">E105</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_16_1"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2_16_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>41</sub></font> = <font color="Maroon">c<sub>36</sub></font>, <font color="Maroon">c<sub>42</sub></font> = <font color="Maroon">c<sub>37</sub></font>, <font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>44</sub></font> = <font color="Maroon">c<sub>26</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_16_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>45</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:92_1_1_1_1_2_16_1_1"/>
( <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font> or  not <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font> )
 ;<br/><a NAME="E2:92_1_1_1_1_2_16_1_1"/><b>then </b>
( ( <font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> ) or ( <font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E14:92_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="chain_1.html#E12:92_1_1_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="chain_1.html#E15:92_1_1_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="chain_1.html#E47:92_1_1_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="chain_1.html#E49:92_1_1_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="chain_1.html#E52:92_1_1_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="chain_1.html#E1:92_1_1_1_1_2_16_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="chain_1.html#T48" target="_self">Th48</a></i>;<br/><b>hence </b><a NAME="E3:92_1_1_1_1_2_16_1_1"/>
( <font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> )
 ;<br/></div><b>end;</b></div><a NAME="E4:92_1_1_1_1_2_16_1"/><b>then </b>
( <font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>44</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E5:92_1_1_1_1_2_16_1"/>
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> or <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E49:92_1_1_1_1_2"><i><font color="Green">E98</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1_2_16_2"><b>suppose </b></a><a NAME="E1:92_1_1_1_1_2_16_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>41</sub></font> = <font color="Maroon">c<sub>36</sub></font>, <font color="Maroon">c<sub>42</sub></font> = <font color="Maroon">c<sub>37</sub></font>, <font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>31</sub></font>, <font color="Maroon">c<sub>44</sub></font> = <font color="Maroon">c<sub>32</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_2_16_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>45</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:92_1_1_1_1_2_16_2_1"/>
( <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font> or  not <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font> )
 ;<br/><a NAME="E2:92_1_1_1_1_2_16_2_1"/><b>then </b>
( ( <font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> ) or ( <font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>45</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E14:92_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="chain_1.html#E16:92_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="chain_1.html#E19:92_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="chain_1.html#E27:92_1_1_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="chain_1.html#E30:92_1_1_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="chain_1.html#E47:92_1_1_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="chain_1.html#E49:92_1_1_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="chain_1.html#E52:92_1_1_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="chain_1.html#E1:92_1_1_1_1_2_16_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="chain_1.html#T48" target="_self">Th48</a></i>;<br/><b>hence </b><a NAME="E3:92_1_1_1_1_2_16_2_1"/>
( <font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>45</sub></font> )
 ;<br/></div><b>end;</b></div><a NAME="E4:92_1_1_1_1_2_16_2"/><b>then </b>
( <font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>44</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E5:92_1_1_1_1_2_16_2"/>
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> or <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E49:92_1_1_1_1_2"><i><font color="Green">E98</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E21:92_1_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="chain_1.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E22:92_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="chain_1.html#T52" target="_self">Th52</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:92_1_1_1"/><b>then </b>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:92_1_1_1"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E5:92_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> st  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> holds <br/> <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:92_1_1_2"/><i><font color="Green">E70</font></i>: 
(  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>thus </b> <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> = 
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="chain_1.html#K8" target="_self">+</a> <span class="p2">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="chain_1.html#T63" target="_self">Th63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="chain_1.html#K8" target="_self">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="chain_1.html#T63" target="_self">Th63</a></i>
<br/>.= 
 <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>

;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>] means  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E6:92_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="chain_1.html#K6" target="_self">0_</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b> <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2">(<span class="default"><a href="chain_1.html#K6" target="_self">0_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K6" target="_self">0_</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="chain_1.html#T61" target="_self">Th61</a></i>
<br/>.= 
 <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="chain_1.html#T61" target="_self">Th61</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E7:92_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> holds  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:92_1_1_4"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; ( ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) or ( not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="chain_1.html#T33" target="_self">Th33</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_1_4_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) or ( not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E2:92_1_1_4"><i><font color="Green">E71</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_4_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) or ( not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 ;<br/><div class="add"><a NAME="E2:92_1_1_4_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><b>hence </b><a NAME="E3:92_1_1_4_1_1"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:92_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="chain_1.html#E2:92_1_1_4"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:92_1_1_4_1_2"/><i><font color="Green">E72</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 ;<br/><div class="add"><a NAME="E2:92_1_1_4_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E2:92_1_1_4"><i><font color="Green">E71</font></i></a>, <a class="ref" href="chain_1.html#T50" target="_self">Th50</a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a> ;<br/><a NAME="E3:92_1_1_4_1_2"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_4_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:92_1_1_4_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:92_1_1_4_1_2_1"/><b>then </b>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:92_1_1_4_1_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E2:92_1_1_4_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:92_1_1_4_1_2_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; ( ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) or ( not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="chain_1.html#T33" target="_self">Th33</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_4_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="E1:92_1_1_4_1_2_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) or ( not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E5:92_1_1_4_1_2_1"><i><font color="Green">E76</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_4_1_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) or ( not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 ;<br/><div class="add"><a NAME="E2:92_1_1_4_1_2_1_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><b>hence </b><a NAME="E3:92_1_1_4_1_2_1_1_1"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:92_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="chain_1.html#E5:92_1_1_4_1_2_1"><i><font color="Green">E76</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:92_1_1_4_1_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:92_1_1_4_1_2_1_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:92_1_1_4_1_2_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="chain_1.html#E5:92_1_1_4_1_2_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="chain_1.html#T50" target="_self">Th50</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:92_1_1_4_1_2"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Olive">b<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="chain_1.html#E5:92_1_1"><i><font color="Green">E69</font></i></a></i>;<br/><a NAME="E5:92_1_1_4_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a> ]
 <b>from </b><i><a class="ref" href="chain_1.html#S4" target="_self">CHAIN_1:sch 4</a>(<a class="txt" href="chain_1.html#E6:92_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="chain_1.html#E3:92_1_1_4_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="chain_1.html#E4:92_1_1_4_1_2"><i><font color="Green">E75</font></i></a>);<br/></i><b>hence </b><a NAME="E6:92_1_1_4_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:92_1_1_4_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="chain_1.html#T64" target="_self">Th64</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E8:92_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>]
 ;<br/><a NAME="E9:92_1_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Olive">b<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="chain_1.html#E5:92_1_1"><i><font color="Green">E69</font></i></a></i>;<br/><b>thus </b><a NAME="E10:92_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>from </b><i><a class="ref" href="chain_1.html#S4" target="_self">CHAIN_1:sch 4</a>(<a class="txt" href="chain_1.html#E6:92_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="chain_1.html#E8:92_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="chain_1.html#E9:92_1_1"><i><font color="Green">E72</font></i></a>);</i><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2"><b>suppose </b></a><a NAME="E1:92_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:92_1_2"/><b>then </b>
 <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="chain_1.html#T53" target="_self">Th53</a></i>;<br/><b>hence </b><a NAME="E3:92_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="chain_1.html#T61" target="_self">Th61</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
