<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="osafree.html#K23" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:80_1_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K23" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:80_1_2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K23" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p5">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/>
<a NAME="E3:80_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="osafree.html#K23" target="_self">OSFreeGen</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p5">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="osafree.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E4:80_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p5">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p5">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:80_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p5">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:80_1_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p5">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:80_1_2_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<a NAME="E4:80_1_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K18">OSNat_Hom</a> <span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="osafree.html#K15" target="_self">LCongruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E1:80_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="osafree.html#E2:80_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="osafree.html#E3:80_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="osafree.html#E1:80_1_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="osafree.html#E3:80_1_2_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:80_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E3:80_1_2_1"><i><font color="Green">E60</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:80_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="osafree.html#E4:80_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
