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<b>let </b><font color="Maroon">U0</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<b>set </b><font color="Maroon">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> where <font color="Olive">o</font> is   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> :  <a href="unialg_1.html#K3">arity</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:11"/>
 <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">U0</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> where <font color="Olive">o</font> is   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> :  <a href="unialg_1.html#K3">arity</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:11_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">U0</font>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a1</font> where <font color="Olive">a1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">U0</font> :  ex <font color="Olive">o</font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> st <br/>(  <a href="unialg_1.html#K3">arity</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">a1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">o</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">U0</font><b> such that </b><br/><a NAME="E4:11_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  ex <font color="Olive">o</font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> st <br/>(  <a href="unialg_1.html#K3">arity</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">o</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">o</font> being   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font><b> such that </b><br/><a NAME="E6:11_1"/><i><font color="Green">E35</font></i>: 
 <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E7:11_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:11_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:11_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> 0 <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font>
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<a NAME="E11:11_1"/><b>then </b>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T113">FINSEQ_2:113</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> ;<br/>
<a NAME="E13:11_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , <a class="ref" href="unialg_1.html#D10">UNIALG_1:def 10</a></i>;<br/>
<a NAME="E14:11_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E15:11_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> where <font color="Olive">o</font> is   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> :  <a href="unialg_1.html#K3">arity</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E2:11"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> where <font color="Olive">o</font> is   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> :  <a href="unialg_1.html#K3">arity</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> 0 </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">U0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:11_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> where <font color="Olive">o</font> is   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> :  <a href="unialg_1.html#K3">arity</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font><b> such that </b><br/><a NAME="E3:11_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">o</font> = 
0 <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font>
<b>by </b><i>, <a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>
;<br/>
<a NAME="E5:11_2"/><b>then </b>
 <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:11_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">U0</font> ;<br/>
<a NAME="E8:11_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a2</font> where <font color="Olive">a2</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">U0</font> :  ex <font color="Olive">o</font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> st <br/>(  <a href="unialg_1.html#K3">arity</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">o</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:11_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">U0</font>
 <b>by </b><i/>;<br/>


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


</div>
