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<b>let </b><font color="Maroon">z</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">N</font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">M</font> =  <a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font>;<br/>
<a NAME="E1:62"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">z</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span><b> such that </b><br/><a NAME="E3:62"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a> &amp; ( for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> iff <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="amistd_1.html#R1" target="_self">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> ) ) &amp;  <a href="amistd_1.html#K5" target="_self">il.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:62"/><i><font color="Green">E39</font></i>: 
the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">f2</font> =  <a href="partfun1.html#K6">id</a> <a href="numbers.html#K5">NAT</a>  as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:62_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E2:62_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="amistd_1.html#K3" target="_self">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:62_1"/>
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="amistd_1.html#K3" target="_self">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E1:62"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:62_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="amistd_1.html#K3" target="_self">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><a NAME="E5:62_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E6:62_1"/><b>then </b>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="amistd_1.html#E3:62"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:62_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:62"/><b>then </b>
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> iff <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="amistd_1.html#R1" target="_self">&lt;=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:62"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="ref" href="amistd_1.html#T3" target="_self">Th3</a>, , </i>;<br/>
<b>hence </b><a NAME="E9:62"/>
 <a href="amistd_1.html#K5" target="_self">il.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="amistd_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>


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