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<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="msualg_6.html#K5" target="_self">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E2:48_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="msualg_6.html#V3" target="_self">invariant</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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/>

<a NAME="E1:48_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</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="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</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="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msualg_6.html#D10" target="_self">Def10</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E2:48_1_1_1"/>
<font color="Maroon">c<sub>6</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</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>4</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>5</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>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:48_1_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:48_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</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>4</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:48_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</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>5</sub></font> )
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:48_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msualg_6.html#E4:48_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:48_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <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#E1:48_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>


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

<b>thus </b><a NAME="E3:48_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="msualg_6.html#V4" target="_self">stable</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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>5</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:48_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</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="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_6.html#D10" target="_self">Def10</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:48_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E3:48_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</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>5</sub></font> )
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:48_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E4:48_1_1_2"/><b>then </b>
<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>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E5:48_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <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#E1:48_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msualg_6.html#E3:48_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a>,<a class="ref" href="msualg_4.html#D5">MSUALG_4:def 5</a><br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="relset_1.html#NM3">Relation</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>4</sub></font>;<br/>



<b>assume </b><a NAME="E4:48_1_1"/>
<font color="Maroon">c<sub>4</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>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>assume </b><a NAME="E6:48_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E7:48_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</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="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_6.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E8:48_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</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>4</sub></font>
 ;<br/>


</div>
