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<b>let </b><font color="Maroon">Z</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">o</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">Z</font>
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font></span>)</span> where <font color="Olive">s'</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> </span>}</span> ;<br/>
<a NAME="E2:24"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">Z</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">o</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font></span>)</span> where <font color="Olive">s'</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> </span>}</span>  <a href="tarski.html#R2">are_equipotent</a>  &amp; not  <a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font></span>)</span> where <font color="Olive">s'</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> </span>}</span>  )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font></span>)</span> where <font color="Olive">s'</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">Z</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<i><font color="Green">E39</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="card_1.html#T81">CARD_1:81</a></i>
;<br/>
<a NAME="E6:24"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font></span>)</span> where <font color="Olive">s'</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">Z</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font></span>)</span> where <font color="Olive">s'</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:24_2_1_1_1_1"/><b>then </b>
 ex <font color="Olive">v'</font> being   <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">v'</font> &amp; <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">v'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ;<br/><b>hence </b><a NAME="E3:24_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">Z</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:24_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:24"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<b>hence </b><a NAME="E8:24"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">o</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


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