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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="modelc_1.html#K58" target="_self">PathChange</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:163_1_1"/><i><font color="Green">E27</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">H<sub>1</sub></font>(<font color="Olive">n</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="163_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:163_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E3:163_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K58" target="_self">PathChange</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font>,<font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">x</font> =  <a href="modelc_1.html#K58" target="_self">PathChange</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font>,<font color="Maroon">n</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="163_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:163_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="163_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:163_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:163_1_1_1_1_1"/><b>then </b>
 <a href="modelc_1.html#K58" target="_self">PathChange</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pai1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:163_1_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E34"><i><font color="Green">Lemma124</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="163_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:163_1_1_1_1_2"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">m</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="modelc_1.html#E35"><i><font color="Green">Lemma125</font></i></a>, <a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/><a NAME="E3:163_1_1_1_1_2"/>
 <a href="modelc_1.html#K58" target="_self">PathChange</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pai2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E35"><i><font color="Green">Lemma125</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:163_1_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E34"><i><font color="Green">Lemma124</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<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="E3:163_1_1"/><i><font color="Green">E53</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>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<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="E4:163_1_1"/>
 <a href="modelc_1.html#K2" target="_self">k_nat</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:163_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>  <a href="modelc_1.html#K58" target="_self">PathChange</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


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