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<b>let </b><font color="Maroon">S</font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">U1</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S</font>;<br/>



<b>reconsider </b><font color="Maroon">C</font> = <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font></span><a href="pboole.html#K10">|]</a></span> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">U1</font> <b>by </b><i><a class="ref" href="msualg_5.html#T20">MSUALG_5:20</a></i>;<br/>
<a NAME="E2:60"/>
<font color="Maroon">C</font> is <a href="osalg_4.html#V1" target="_self">os-compatible</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_4.html#D1" target="_self">OSALG_4:def 1</a><br/>

<b>assume </b><a NAME="E1:60_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">s3</font> = <font color="Maroon">s1</font>, <font color="Maroon">s4</font> = <font color="Maroon">s2</font> as   <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="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:60_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">O1</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> as   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>
<a NAME="E5:60_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">O1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s4</font>
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#D18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E6:60_1"/>
( <font color="Maroon">C</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">C</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/>
<b>hence </b><a NAME="E7:60_1"/>
( <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">C</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s1</font> iff <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">C</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:60"/>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font></span><a href="pboole.html#K10">|]</a></span> is   <a href="osalg_4.html#NM4">OSCongruence</a> of <font color="Maroon">U1</font>
 <b>by </b><i/>;<br/>


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