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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="osalg_4.html#V2" target="_self">locally_directed</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:56"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="osalg_3.html#V1">order-sorted</a> )
 ;<br/>

<a NAME="E2:56"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="msualg_1.html#K3">Args</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:56_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<i><font color="Green">E34</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K15" target="_self">OSQuotCharact</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="msualg_4.html#K5">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="osalg_4.html#K14" target="_self">OSQuotCharact</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="osalg_4.html#D21" target="_self">Def21</a></i>
;<br/>
<a NAME="E4:56_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <span class="p4">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:56_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="osalg_4.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E7:56_1"/><i><font color="Green">E36</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:56_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E9:56_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E37</font></i>: <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
;<br/>
<a NAME="E11:56_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <b>by </b><i><a class="txt" href="osalg_4.html#E10:56_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/>
<a NAME="E13:56_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="osalg_4.html#K20" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E14:56_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E7:56_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="osalg_4.html#E10:56_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="osalg_4.html#E11:56_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:56_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <span class="p3">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E16:56_1"/><i><font color="Green">E41</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:56_1_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:56_1_3"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E1:56_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:56_1_3"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_4.html#E1:56_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> as   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font><b> such that </b><br/><a NAME="E7:56_1_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E6:56_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="osalg_4.html#E1:56_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="osalg_4.html#E3:56_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="osalg_4.html#D15" target="_self">Def15</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E16:56_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="osalg_4.html#E1:56_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="osalg_4.html#E3:56_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_4.html#K20" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E7:56_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="osalg_4.html#D27" target="_self">Def27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self">Def26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="osalg_4.html#E16:56_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="osalg_4.html#E1:56_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="osalg_4.html#E3:56_1_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="osalg_4.html#E7:56_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
;<br/><b>hence </b><a NAME="E9:56_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E18:56_1"/>
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E19:56_1"/><b>then </b>
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <span class="p3">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="msualg_4.html#K6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <span class="p4">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="msafree.html#T1">MSAFREE:1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <span class="p3">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="osalg_4.html#E3:56_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="osalg_4.html#E6:56_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="osalg_4.html#D20" target="_self">Def20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="osalg_4.html#E7:56_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="osalg_4.html#E15:56_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="osalg_4.html#D16" target="_self">Def16</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="txt" href="osalg_4.html#E13:56_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self">Def26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E22:56_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_4.html#E16:56_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="osalg_4.html#E17:56_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:56"/>
 <a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a><br/>

<a NAME="E4:56"/>
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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><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>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<b>assume </b><a NAME="E1:56_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:56_2"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K20" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E4:56_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</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#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:56_2_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</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#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<a NAME="E2:56_2_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E3:56_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:56_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K6" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K6" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="osalg_4.html#D13" target="_self">Def13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:56_2_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="osalg_4.html#D12" target="_self">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:56_2_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E3:56_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="osalg_4.html#D12" target="_self">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="osalg_4.html#E5:56_2_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="osalg_4.html#E7:56_2_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E10:56_2_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E5:56_2_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="osalg_4.html#E7:56_2_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E11:56_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</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#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</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#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="txt" href="osalg_4.html#E3:56_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self">Def26</a></i>;<br/>
<a NAME="E12:56_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K17">MSCng</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56_2_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="osalg_4.html#E10:56_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="msualg_4.html#D19">MSUALG_4:def 19</a></i>;<br/>
<a NAME="E13:56_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msualg_4.html#D20">MSUALG_4:def 20</a></i>;<br/>
<a NAME="E14:56_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="ref" href="osalg_4.html#D25" target="_self">Def25</a></i>;<br/>
<b>hence </b><a NAME="E15:56_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E10:56_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="osalg_4.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:56_2"/>
<span class="p1">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:56"/>
 <a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font> is <a href="msualg_3.html#V1">"1-1"</a>
 <b>by </b><i><a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/>

<b>thus </b><a NAME="E6:56"/>
 <a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font> is <a href="osalg_3.html#V1">order-sorted</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_3.html#D1">OSALG_3:def 1</a><br/>

<b>assume </b><a NAME="E1:56_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:56_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> as   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>
<a NAME="E5:56_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="osalg_1.html#D18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E6:56_3"/><i><font color="Green">E37</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:56_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="osalg_1.html#D18">OSALG_1:def 18</a></i>;<br/>
<b>hence </b><a NAME="E8:56_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_4.html#E2:56_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="osalg_4.html#E6:56_3"><i><font color="Green">E37</font></i></a></i>;<br/>

<a NAME="E9:56_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:56_3"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E10:56_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K6" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D13" target="_self">Def13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:56_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E13:56_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="osalg_4.html#D12" target="_self">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</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="osalg_4.html#E12:56_3"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="osalg_4.html#E5:56_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="osalg_4.html#E12:56_3"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E17:56_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E12:56_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:56_3"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="txt" href="osalg_4.html#E1:56_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="osalg_3.html#D1">OSALG_3:def 1</a></i>;<br/>
<a NAME="E19:56_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E13:56_3"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E20:56_3"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="osalg_4.html#T14" target="_self">Th14</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K20" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E19:56_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="osalg_4.html#D27" target="_self">Def27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="txt" href="osalg_4.html#E18:56_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self">Def26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_4.html#K20" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green">E33</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self">Def26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="osalg_4.html#E20:56_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="osalg_4.html#D27" target="_self">Def27</a></i>
;<br/>


</div><b>end;</b></div>


</div>
