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<div class="add">

<b>set </b><font color="Maroon">f</font> =  <a href="pralg_1.html#K7" target="_self">TrivialOp</a> <font color="Maroon">k</font>;<br/>
<a NAME="E1:15_1_1"/><i><font color="Green">E23</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K7" target="_self">TrivialOp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K7" target="_self">TrivialOp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:15_1_1"/>
 <a href="pralg_1.html#K7" target="_self">TrivialOp</a> <font color="Maroon">k</font> is <a href="unialg_1.html#V1">homogeneous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D1">UNIALG_1:def 1</a><br/>

<b>assume </b><a NAME="E1:15_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K7" target="_self">TrivialOp</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K7" target="_self">TrivialOp</a> <font color="Maroon">k</font></span>)</span> )
 ;<br/>

<a NAME="E2:15_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 ;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_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 <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>;<br/><b>assume </b><a NAME="E1:15_1_1_2"/><i><font color="Green">E26</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K7" target="_self">TrivialOp</a> <font color="Maroon">k</font></span>)</span> )
 ;<br/><a NAME="E2:15_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:15_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span>
 ;<br/><a NAME="E2:15_1_1_2_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E3:15_1_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:15_1_1_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>hence </b><a NAME="E5:15_1_1_2_1"/>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K7" target="_self">TrivialOp</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15_1_1"/>
(  <a href="pralg_1.html#K7" target="_self">TrivialOp</a> <font color="Maroon">k</font> is <a href="unialg_1.html#V2">quasi_total</a> &amp; not  <a href="pralg_1.html#K7" target="_self">TrivialOp</a> <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i>, <a class="ref" href="unialg_1.html#T1">UNIALG_1:1</a>, <a class="ref" href="unialg_1.html#D2">UNIALG_1:def 2</a></i>;<br/>


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