<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green">E40</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E3:44"/>
(  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E4:44"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<b>then </b><i><font color="Green">E45</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i/>
<br/>.= 
2
<b>by </b><i>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>
;<br/>
<a NAME="E6:44"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:44"/><b>then </b><i><font color="Green">E50</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E8:44"/><b>then </b><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:44"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_1"/>
( ( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) or ( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> ) or ( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> ) or ( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_1"><b>suppose </b></a><a NAME="E1:44_2_1_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_2_1_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_2"><b>suppose </b></a><a NAME="E1:44_2_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_2_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_3"><b>suppose </b></a><a NAME="E1:44_2_1_3"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_2_1_3"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_4"><b>suppose </b></a><a NAME="E1:44_2_1_4"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_2_1_4"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:44"/>
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>


</div>
