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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">X</font> be   <a href="struct_0.html#NM8">ManySortedSet</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="msafree.html#K3">coprod</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> ;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>,   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="msafree.html#K3">coprod</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> ] <b>means </b>(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a<sub>2</sub></font> holds <br/>( ( <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> implies for <font color="Olive">o1</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> holds <br/> <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o1</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> ) &amp; ( <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3">coprod</a> <font color="Maroon">X</font></span>)</span> implies  ex <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">i</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> &amp; <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K2">coprod</a> <font color="Olive">i</font>,<font color="Maroon">X</font> ) ) ) ) );<br/>
<b>set </b><font color="Maroon">a</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E1:6"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:6"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3">coprod</a> <font color="Maroon">X</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>thus </b><a NAME="E4:6"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K1" target="_self">OSREL</a> <font color="Maroon">X</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o</font>,<font color="Maroon">b</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K1" target="_self">OSREL</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:6_1"/><b>then </b>
for <font color="Olive">o1</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o1</font>,<font color="Maroon">b</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:6_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o</font>,<font color="Maroon">b</font>]
 ;<br/>


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

<b>assume </b><a NAME="E5:6"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o</font>,<font color="Maroon">b</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o1</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:6_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:6_2"/><b>then </b>
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:6_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o1</font>,<font color="Maroon">b</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K1" target="_self">OSREL</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div>
