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<div class="add">

<a NAME="E1:161_1_2"/>
for <font color="Olive">pai1</font>, <font color="Olive">pai2</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font>  st ( 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="Olive">pai1</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> ) &amp; ( 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="Olive">pai2</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> ) holds <br/><font color="Olive">pai1</font> <a href="funct_2.html#R2">=</a> <font color="Olive">pai2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_2_1"><b>proof </b></a><div class="add">

<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>assume </b><b>that </b><br/><a NAME="E1:161_1_2_1"/><i><font color="Green">E53</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">pai1</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>and </b><br/><a NAME="E2:161_1_2_1"/><i><font color="Green">E54</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">pai2</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>
 ;<br/>

<a NAME="E3:161_1_2_1"/>
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">pai1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pai2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:161_1_2_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<font color="Maroon">pai1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">pai2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:161_1_2_1_1"/>
<font color="Maroon">pai1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pai2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:161_1_2_1"/>
<font color="Maroon">pai1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">pai2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:161_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font>  st ( 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="Olive">b<sub>1</sub></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> ) &amp; ( 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="Olive">b<sub>2</sub></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> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>


</div>
