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

<b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ;<br/><b>let </b><font color="Maroon" title="c3">B</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="polyalg1.html#L1" title="POLYALG1:struct.1">AlgebraStr</a> of <font color="Maroon" title="c2">L</font>;<br/><b>let </b><font color="Maroon" title="c4">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">B</font>;<br/><b>let </b><font color="Maroon" title="c5">X</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">B</font>;<br/>







<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E23">A1</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> iff ( <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font> &amp;  ex <font color="Olive" title="b2">C</font> being   <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font> st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) ) )
 ;<br/>

<a NAME="E2:44"/>
<font color="Maroon" title="c3">B</font> is    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="polyalg1.html#T11" target="_self" title="POLYALG1:th.11">Th11</a></i>;<br/>
<a NAME="E3:44"/><b>then </b><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="polyalg1.html#E1:44"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:44"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font> is <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="polyalg1.html#D4" target="_self" title="POLYALG1:def.4">POLYALG1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:44_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_4.html#D1" title="VECTSP_4:def.1">VECTSP_4:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>;<br/>

<b>assume </b><a NAME="E1:44_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:44_1_1_1"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">C</font> being    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font><b> such that </b><br/><a NAME="E3:44_1_1_1"/><i><font color="Green" title="E27">A5</font></i>: 
( <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> )
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:44"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:44_1_1_1"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c9">C</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="polyalg1.html#E3:44_1_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="polyalg1.html#E5:44_1_1_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">x'</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c13">y'</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> ;<br/><b>reconsider </b><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c12">x'</font>, <font color="Maroon" title="c15">y1</font> = <font color="Maroon" title="c13">y'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">C</font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:44_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="polyalg1.html#E1:44_1_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="polyalg1.html#E3:44_1_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><a NAME="E9:44_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">y1</font>
 <b>by </b><i><a class="ref" href="polyalg1.html#T15" target="_self" title="POLYALG1:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E10:44_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:44_1_1_1"><i><font color="Green" title="E27">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:44_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:44"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:44_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font> holds <br/><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>;<br/><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>;<br/>



<b>assume </b><a NAME="E1:44_1_2"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:44_1_2_1"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">C</font> being    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font><b> such that </b><br/><a NAME="E3:44_1_2_1"/><i><font color="Green" title="E27">A9</font></i>: 
( <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> )
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:44"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:44_1_2_1"/><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c9">C</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="polyalg1.html#E3:44_1_2_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="polyalg1.html#E5:44_1_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">v'</font> = <font color="Maroon" title="c7">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">C</font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:44_1_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="polyalg1.html#E1:44_1_2_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="polyalg1.html#E3:44_1_2_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><a NAME="E8:44_1_2_1"/>
<font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">v'</font>
 <b>by </b><i><a class="ref" href="polyalg1.html#T17" target="_self" title="POLYALG1:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E9:44_1_2_1"/>
<font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:44_1_2_1"><i><font color="Green" title="E27">A9</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:44_1_2"/>
<font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:44"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


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


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

<b>thus </b><a NAME="E5:44"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</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" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>;<br/>

<b>assume </b><a NAME="E1:44_2"/><i><font color="Green" title="E25">A11</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:44_2_1"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">C</font> being    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font><b> such that </b><br/><a NAME="E3:44_2_1"/><i><font color="Green" title="E27">A13</font></i>: 
( <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> )
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:44"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:44_2_1"/><i><font color="Green" title="E28">A14</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c9">C</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="polyalg1.html#E3:44_2_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="polyalg1.html#E5:44_2_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">x'</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c13">y'</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> ;<br/><b>reconsider </b><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c12">x'</font>, <font color="Maroon" title="c15">y1</font> = <font color="Maroon" title="c13">y'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">C</font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:44_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="polyalg1.html#E1:44_2_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="polyalg1.html#E3:44_2_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><a NAME="E9:44_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">y1</font>
 <b>by </b><i><a class="ref" href="polyalg1.html#T16" target="_self" title="POLYALG1:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E10:44_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:44_2_1"><i><font color="Green" title="E27">A13</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:44_2"/>
<font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:44"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:44_3"/>
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">C</font> being    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font><b> such that </b><br/><a NAME="E3:44_3"/><i><font color="Green" title="E25">A15</font></i>: 
( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Y</font> )
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:44"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:44_3"/><i><font color="Green" title="E26">A16</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c7">C</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="polyalg1.html#E3:44_3"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="polyalg1.html#E5:44_3"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><a NAME="E7:44_3"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E8:44_3"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:44_3"><i><font color="Green" title="E25">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:44"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:44"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:44_4"/>
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">C</font> being    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font><b> such that </b><br/><a NAME="E3:44_4"/><i><font color="Green" title="E25">A17</font></i>: 
( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Y</font> )
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:44"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:44_4"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c7">C</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="polyalg1.html#E3:44_4"><i><font color="Green" title="E25">A17</font></i></a>, <a class="txt" href="polyalg1.html#E5:44_4"><i><font color="Green" title="E26">A18</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><a NAME="E7:44_4"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E8:44_4"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:44_4"><i><font color="Green" title="E25">A17</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:44"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:44"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div>
