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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:79"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:79"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="msualg_4.html#V2">MSEquivalence_Relation-like</a>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E3:79"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="msualg_6.html#V4" target="_self">stable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="msualg_6.html#M1" target="_self">Endomorphism</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_6.html#D9" target="_self">MSUALG_6:def 9</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:79_1"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> ;<br/>
<a NAME="E4:79_1"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:79"><i><font color="Green">E54</font></i></a>, <a class="txt" href="msualg_6.html#E1:79_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E5:79_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T45" target="_self">Th45</a></i>;<br/>
<b>hence </b><a NAME="E6:79_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"><font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:79"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:79"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="msualg_6.html#V3" target="_self">invariant</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_6.html#D8" target="_self">MSUALG_6:def 8</a><br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:79_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</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>5</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> <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="ref" href="msualg_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:79_2"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:79_2"/>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:79"><i><font color="Green">E54</font></i></a>, <a class="txt" href="msualg_6.html#E3:79_2"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E6:79_2"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:79_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="msualg_6.html#T47" target="_self">Th47</a></i>;<br/>
<b>hence </b><a NAME="E7:79_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:79"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:79"/>
<font color="Maroon">c<sub>4</sub></font> is   <a href="msualg_6.html#NM4">EquationalTheory</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:79"><i><font color="Green">E55</font></i></a>, <a class="txt" href="msualg_6.html#E3:79"><i><font color="Green">E56</font></i></a>, <a class="ref" href="msualg_4.html#D5">MSUALG_4:def 5</a></i>;<br/>


</div>
