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<div class="add">

<b>consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <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> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">A</font> <b>by </b><i><a class="ref" href="unialg_1.html#T3">UNIALG_1:3</a></i>;<br/>
<b>set </b><font color="Maroon">U0</font> =  <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">A</font>,<span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="unialg_1.html#K1">*&gt;</a></span> #);<br/>
<a NAME="E4:33_1_1"/><i><font color="Green">E20</font></i>: 
( the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">A</font>,<span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="unialg_1.html#K1">*&gt;</a></span> #) is <a href="unialg_1.html#V5">quasi_total</a> &amp; the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">A</font>,<span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="unialg_1.html#K1">*&gt;</a></span> #) is <a href="unialg_1.html#V4">homogeneous</a> &amp; the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">A</font>,<span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="unialg_1.html#K1">*&gt;</a></span> #) is <a href="relat_1.html#V2">non-empty</a> )
 
<b>by </b><i><a class="ref" href="unialg_1.html#T4">UNIALG_1:4</a></i>;<br/>
<a NAME="E5:33_1_1"/>
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">A</font>,<span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="unialg_1.html#K1">*&gt;</a></span> #) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_1_1"/><i><font color="Green">E26</font></i>: 
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">A</font>,<span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="unialg_1.html#K1">*&gt;</a></span> #) <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:33_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">A</font>,<span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="unialg_1.html#K1">*&gt;</a></span> #) <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">U0</font> =  <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">A</font>,<span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="unialg_1.html#K1">*&gt;</a></span> #) as   <a href="unialg_1.html#NM2">Universal_Algebra</a> <b>by </b><i>, <a class="ref" href="unialg_1.html#D7">UNIALG_1:def 7</a>, <a class="ref" href="unialg_1.html#D8">UNIALG_1:def 8</a>, <a class="ref" href="unialg_1.html#D9">UNIALG_1:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">U0</font>
;<br/>

<a NAME="E7:33_1_1"/>
(  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="unialg_1.html#K1">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> 1 as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> <b>by </b><i/>;<br/>
<a NAME="E9:33_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E10:33_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:33_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E2:33_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:33_1_1"/><b>then </b>
 <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</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/>
<b>hence </b><a NAME="E13:33_1_1"/>
( <font color="Maroon">U0</font> is <a href="unialg_2.html#V2" target="_self">with_const_op</a> &amp; <font color="Maroon">U0</font> is <a href="unialg_1.html#V3">strict</a> )
 <b>by </b><i><a class="txt" href="unialg_2.html#E4:33_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>


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