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<b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be  <a href="partfun1.html#V1">total</a>  <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">pai1</font> be    <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font>, <font color="Maroon">pai2</font> be    <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font>;<br/><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/>







<b>assume </b><a NAME="E1:164"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">pai1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pai2</font> <a href="funct_2.html#K3">.</a> 0
 ;<br/>

<b>set </b><font color="Maroon">pai</font> =  <a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font>;<br/>
<a NAME="E2:164"/>
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"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <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="164_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/>

<b>set </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:164_1_1"/>
( <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> or <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1_1"><b>suppose </b></a><a NAME="E1:164_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><i><font color="Green">E50</font></i>: <span class="p1">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <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> = 
 <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"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">pai1</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>
<b>by </b><i><a class="txt" href="modelc_1.html#E35"><i><font color="Green">Lemma125</font></i></a>, </i>
;<br/><a NAME="E3:164_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <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#T38">NAT_1:38</a></i>;<br/><span class="p1">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</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>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">pai1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="modelc_1.html#E37"><i><font color="Green">Lemma127</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E5:164_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <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>, <a class="txt" href="modelc_1.html#E36"><i><font color="Green">Lemma126</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1_2"><b>suppose </b></a><a NAME="E1:164_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <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> = 
 <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"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">pai2</font> <a href="funct_1.html#K1">.</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#K10">-</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E35"><i><font color="Green">Lemma125</font></i></a>, </i>
<br/>.= 
<font color="Maroon">pai1</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>
<b>by </b><i><a class="txt" href="modelc_1.html#E34"><i><font color="Green">Lemma124</font></i></a>, <a class="txt" href="modelc_1.html#E35"><i><font color="Green">Lemma125</font></i></a></i>
;<br/><a NAME="E3:164_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="modelc_1.html#E35"><i><font color="Green">Lemma125</font></i></a></i>;<br/><span class="p1">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</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>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">pai1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="modelc_1.html#E37"><i><font color="Green">Lemma127</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E5:164_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <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>, <a class="txt" href="modelc_1.html#E36"><i><font color="Green">Lemma126</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_1_1_3"><b>suppose </b></a><a NAME="E1:164_1_1_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:164_1_1_3"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then 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="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/><i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</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>
<b>by </b><i/>
<br/>.= 
<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#E36"><i><font color="Green">Lemma126</font></i></a>, </i>
;<br/><span class="p1">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <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> = 
 <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"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">pai2</font> <a href="funct_1.html#K1">.</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#K10">-</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E35"><i><font color="Green">Lemma125</font></i></a>, </i>
<br/>.= 
<font color="Maroon">pai2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>

;<br/><b>hence </b><a NAME="E6:164_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font></span>)</span> <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>, <a class="txt" href="modelc_1.html#E37"><i><font color="Green">Lemma127</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:164"/>
 <a href="modelc_1.html#K59" target="_self">PathConc</a> <font color="Maroon">pai1</font>,<font color="Maroon">pai2</font>,<font color="Maroon">k</font> is    <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div>
