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<div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> = <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>;<br/>
<a NAME="E1:161_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
;<br/>
<b>consider </b><font color="Maroon">IT</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:161_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>();<br/></i>
<a NAME="E4:161_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">IT</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:161_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:161_1_1_1"/>
<font color="Maroon">IT</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:161_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">IT</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">IT</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>set </b><font color="Maroon">n2</font> = <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font>;<br/>
<a NAME="E1:161_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">IT</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<font color="Maroon">IT</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>

;<br/>
<b>hence </b><a NAME="E3:161_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">IT</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">IT</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as    <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<b>thus </b><a NAME="E7:161_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">IT</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
