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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> $1</span>)</span>;<br/>
<a NAME="E1:163_1_1"/><i><font color="Green" title="E122">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="c6">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<b>assume </b><a NAME="E1:163_1_1_1"/><i><font color="Green" title="E123">A2</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:163_1_1_1"><i><font color="Green" title="E123">A2</font></i></a></i>;<br/>
<a NAME="E3:163_1_1_1"/><i><font color="Green" title="E124">A3</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">x</font> =  <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c7">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" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> or  not <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">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" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font></span><br/><div class="add"><a NAME="E2:163_1_1_1_1_1"/><b>then </b>
 <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">pai1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</a></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" title="c6">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="#E3:163_1_1_1"><i><font color="Green" title="E124">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E125">A4</font></i>: 
not <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">k</font>;<br/><b>reconsider </b><font color="Maroon" title="c10">m</font> = <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:163_1_1_1_1_2"><i><font color="Green" title="E125">A4</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/><a NAME="E3:163_1_1_1_1_2"/>
 <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pai2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="#E1:163_1_1_1_1_2"><i><font color="Green" title="E125">A4</font></i></a>, <a class="ref" href="modelc_1.html#D68" target="_self" title="MODELC_1:def.68">Def68</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" title="c6">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="#E3:163_1_1_1"><i><font color="Green" title="E124">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">IT</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:163_1_1"/><i><font color="Green" title="E123">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c6">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E1:163_1_1"><i><font color="Green" title="E122">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c6">IT</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<font color="Olive" title="b1">n</font></span><br/>

<b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c7">n</font></span><br/>

<a NAME="E4:163_1_1"/>
 <a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:163_1_1"/>
<font color="Maroon" title="c6">IT</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K58" title="MODELC_1:func.58">PathChange</a> <font color="Maroon" title="c3">pai1</font>,<font color="Maroon" title="c4">pai2</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="#E3:163_1_1"><i><font color="Green" title="E123">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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