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<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="prelamb.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>2</sub></font></span><a href="prelamb.html#K6" target="_self">]</a></span>,0</span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E2:17_1_1"/><i><font color="Green">E5</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>2</sub></font></span><a href="prelamb.html#K6" target="_self">]</a></span>,0</span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>2</sub></font></span><a href="prelamb.html#K6" target="_self">]</a></span>,0</span><a href="tarski.html#K4">]</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a> <b>by </b><i><a class="txt" href="prelamb.html#E5"><i><font color="Green">Lemma2</font></i></a>, <a class="ref" href="trees_1.html#T48">TREES_1:48</a>, <a class="ref" href="trees_2.html#D8">TREES_2:def 8</a></i>;<br/>
<a NAME="E4:17_1_1"/><i><font color="Green">E6</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>2</sub></font></span><a href="prelamb.html#K6" target="_self">]</a></span>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:17_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>2</sub></font></span><a href="prelamb.html#K6" target="_self">]</a></span>,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E6:17_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="prelamb.html#E4:17_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="prelamb.html#NM2">PreProof</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="trees_2.html#D9">TREES_2:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E8:17_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

<a NAME="E9:17_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="prelamb.html#E2:17_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><i><font color="Green">E8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="trees_2.html#K2">-level</a> 1;<br/><b>assume </b><a NAME="E2:17_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="trees_2.html#K2">-level</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:17_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="trees_2.html#K2">-level</a> 1
 ;<br/><a NAME="E4:17_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E6:17_1_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><a NAME="E7:17_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="prelamb.html#E2:17_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E8:17_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="prelamb.html#E6:17_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green">E9</font></i>:  <a href="trees_2.html#K10">branchdeg</a> <font color="Maroon">c<sub>6</sub></font> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="trees_2.html#D13">TREES_2:def 13</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="prelamb.html#E9:17_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="prelamb.html#E10:17_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="trees_2.html#T15">TREES_2:15</a></i>
;<br/>
<a NAME="E12:17_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>2</sub></font></span><a href="prelamb.html#K6" target="_self">]</a></span>,0</span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="prelamb.html#E2:17_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E13:17_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="prelamb.html#K6" target="_self">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>2</sub></font></span><a href="prelamb.html#K6" target="_self">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E14:17_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="prelamb.html#V2" target="_self">correct</a>
 <b>by </b><i><a class="txt" href="prelamb.html#E11:17_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="prelamb.html#D5" target="_self">Def5</a></i>;<br/>


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