<?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="group_1.html#L1">HGrStr</a> ;<br/><b>let </b><font color="Maroon">B</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="polyalg1.html#L1" target="_self">AlgebraStr</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">B</font>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> is <a href="polyalg1.html#V3" target="_self">opers_closed</a>
 ;<br/>

<a NAME="E2:44"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> <a href="realset1.html#K1">||</a> <font color="Maroon">A</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E4:44"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<a NAME="E1:44_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:44_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>

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

<b>assume </b><a NAME="E3:44_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:44_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</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/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font>, <font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/>
<a NAME="E7:44_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:44_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E9:44_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ad</font> = <font color="Maroon">f1</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f2</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">B</font> <a href="realset1.html#K1">||</a> <font color="Maroon">A</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E7:44"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:44"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E9:44"/>
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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:44_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</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/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font>, <font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/>
<a NAME="E5:44_2"/>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E6:44_2"/>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">mu</font> = <font color="Maroon">f2</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f4</font> = the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">B</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E12:44"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E13:44"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:44"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E15:44"/>
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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:44_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">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="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_3"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</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/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/>
<a NAME="E6:44_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">A</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:44_3"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:44_3"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">lm</font> = <font color="Maroon">f4</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E17:44"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E19:44"/>
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/>
<b>set </b><font color="Maroon">c</font> =  <a href="polyalg1.html#G1" target="_self">AlgebraStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">ad</font>,<font color="Maroon">mu</font>,<font color="Maroon">z</font>,<font color="Maroon">u</font>,<font color="Maroon">lm</font> #);<br/>
<a NAME="E21:44"/><i><font color="Green">E60</font></i>: 
 <a href="vectsp_1.html#K2">1_</a> <a href="polyalg1.html#G1" target="_self">AlgebraStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">ad</font>,<font color="Maroon">mu</font>,<font color="Maroon">z</font>,<font color="Maroon">u</font>,<font color="Maroon">lm</font> #) <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">B</font>
 
;<br/>
<a NAME="E22:44"/>
 <a href="rlvect_1.html#K1">0.</a> <a href="polyalg1.html#G1" target="_self">AlgebraStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">ad</font>,<font color="Maroon">mu</font>,<font color="Maroon">z</font>,<font color="Maroon">u</font>,<font color="Maroon">lm</font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">B</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">C</font> =  <a href="polyalg1.html#G1" target="_self">AlgebraStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">ad</font>,<font color="Maroon">mu</font>,<font color="Maroon">z</font>,<font color="Maroon">u</font>,<font color="Maroon">lm</font> #) as  <a href="polyalg1.html#V1" target="_self">strict</a>  <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E24:44"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/>


</div>
