<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">L</font>;<br/>



<b>set </b><font color="Maroon">n</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/>
<a NAME="E1:14"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of  <a href="xboole_0.html#K1">{}</a>  holds  <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of  <a href="xboole_0.html#K1">{}</a> ;<br/>

<a NAME="E1:14_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E2:14_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#T55">POLYNOM1:55</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:14"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM5">Function</a> of  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="funct_2.html#NM1">Function</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#T55">POLYNOM1:55</a></i>;<br/>
<i><font color="Green">E32</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> = 
<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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i/>
;<br/>
<b>set </b><font color="Maroon">a</font> = <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> ;<br/>
<b>take </b>
<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> 
;<br/>

<a NAME="E6:14"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:14_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1</font> as   <a href="polynom1.html#NM1">bag</a> of  <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="E6:14_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>then </b><font color="Maroon">p2</font> = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E8:14_3"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:14_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:14"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_4"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">u1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">u2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_4"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">u2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E4:14_4"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</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/>
<font color="Maroon">u2</font> = 
<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</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/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E6:14_4"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:14"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E9:14"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#D2">FUNCOP_1:def 2</a></i>;<br/>


</div>
