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<b>let </b><font color="Maroon">S</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">A</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">C1</font> be  <a href="msualg_4.html#V3">MSEquivalence-like</a> <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">A</font>, <font color="Maroon">C2</font> be  <a href="msualg_4.html#V3">MSEquivalence-like</a> <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">A</font>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">C1</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">C2</font>
 ;<br/>

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/>



<b>assume </b><a NAME="E2:44"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E3:44"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C2</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E4:44"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E5:44"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<b>thus </b><a NAME="E6:44"/>
 <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C2</font>,<font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">x</font>
 
;<br/>
<a NAME="E3:44_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E4:44_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<a NAME="E5:44_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>hence </b><a NAME="E6:44_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C2</font>,<font color="Maroon">y</font>
 ;<br/>


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

<b>assume </b><a NAME="E7:44"/>
<font color="Maroon">A</font> is <a href="msualg_1.html#V5">non-empty</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font> ;<br/>
<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E9:44"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">y</font>
 ;<br/>

<a NAME="E10:44"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E11:44"/><b>then </b>
<font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font> <a href="relat_2.html#R3">is_symmetric_in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>
<a NAME="E12:44"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>
<a NAME="E13:44"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:44"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E15:44"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<a NAME="E16:44"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>hence </b><a NAME="E17:44"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C2</font>,<font color="Maroon">x</font>
 ;<br/>


</div>
