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<b>let </b><font color="Maroon" title="c1">S1</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">M</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>assume </b><a NAME="E2:20"/>
not  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c2">M</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:20"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>
 <b>and </b><br/><a NAME="E5:20"/><i><font color="Green" title="E14">A3</font></i>: 
not <span class="p1">(<span class="default"><a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:20"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>and </b><br/><a NAME="E8:20"/><i><font color="Green" title="E16">A5</font></i>: 
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E5:20"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c4">i</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="txt" href="osalg_2.html#E4:20"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/>
<a NAME="E10:20"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s2</font></span>)</span> where <font color="Olive" title="b1">s2</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">s</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="osalg_2.html#D4" target="_self" title="OSALG_2:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:20"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X1</font>
 <b>and </b><br/><a NAME="E13:20"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c7">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s2</font></span>)</span> where <font color="Olive" title="b1">s2</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">s</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="osalg_2.html#E7:20"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">s1</font> being   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font><b> such that </b><br/><a NAME="E15:20"/><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c7">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font>
 <b>and </b><br/><a NAME="E16:20"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c8">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E13:20"><i><font color="Green" title="E18">A7</font></i></a></i>;<br/>
<a NAME="E17:20"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font>
 
<b>by </b><i><a class="txt" href="osalg_2.html#E1:20"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E18:20"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="txt" href="osalg_2.html#E16:20"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E19:20"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font>
 
<b>by </b><i><a class="txt" href="osalg_2.html#E12:20"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="osalg_2.html#E15:20"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="osalg_2.html#E17:20"><i><font color="Green" title="E21">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:20"/>
contradiction
 <b>by </b><i><a class="txt" href="osalg_2.html#E8:20"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="osalg_2.html#E18:20"><i><font color="Green" title="E22">A11</font></i></a></i>;<br/>


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