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<b>let </b><font color="Maroon">S1</font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">M</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">A</font>
 ;<br/>

<b>assume </b><a NAME="E2:20"/>
not  <a href="osalg_2.html#K2" target="_self">OSCl</a> <font color="Maroon">M</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:20"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>
 <b>and </b><br/><a NAME="E5:20"/><i><font color="Green">E44</font></i>: 
not <span class="p1">(<span class="default"><a href="osalg_2.html#K2" target="_self">OSCl</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <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="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:20"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K2" target="_self">OSCl</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E8:20"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> <b>by </b><i/>;<br/>
<a NAME="E10:20"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K2" target="_self">OSCl</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font></span>)</span> where <font color="Olive">s2</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:20"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>and </b><br/><a NAME="E13:20"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font></span>)</span> where <font color="Olive">s2</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">s1</font> being   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font><b> such that </b><br/><a NAME="E15:20"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E16:20"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>
<a NAME="E17:20"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E18:20"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#D18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E19:20"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, <a class="ref" href="osalg_2.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E20:20"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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