<?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  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">B</font>;<br/>







<b>assume </b><a NAME="E1:45"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp;  ex <font color="Olive">C</font> being   <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> ) ) )
 ;<br/>

<a NAME="E2:45"/>
<font color="Maroon">B</font> is    <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:45"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:45"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="polyalg1.html#D4" target="_self">POLYALG1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:45_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being    <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:45_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:45_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> <b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> ;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x'</font>, <font color="Maroon">y1</font> = <font color="Maroon">y'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> <b>by </b><i>, , , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E9:45_1_1_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:45_1_1_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:45_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><br/> for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> holds <br/><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>



<b>assume </b><a NAME="E1:45_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being    <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:45_1_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:45_1_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> <b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> <b>by </b><i>, , , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E8:45_1_2_1"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:45_1_2_1"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1_2"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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


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

<b>thus </b><a NAME="E5:45"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E1:45_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being    <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:45_2_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:45_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> <b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> ;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x'</font>, <font color="Maroon">y1</font> = <font color="Maroon">y'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> <b>by </b><i>, , , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E9:45_2_1"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:45_2_1"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_2"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being    <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:45_3"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:45_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> <b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E7:45_3"/>
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:45_3"/>
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:45"/>
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_4"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being    <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:45_4"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:45_4"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> <b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E7:45_4"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:45_4"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:45"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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