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<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="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>;<br/>





<b>take </b><font color="Maroon" title="c5">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">C</font> = <font color="Maroon" title="c5">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/>
<a NAME="E2:42"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c6">C</font> is <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_1"/>
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> holds <br/><font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</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="42_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">u</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="E1:42_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">u</font>, <font color="Maroon" title="c10">y</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="c4">A</font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:42_1_1"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>
<a NAME="E3:42_1_1"/>
<font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">x</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="E4:42_1_1"/>
<font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:42_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> <font color="Maroon" title="c6">C</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> <font color="Maroon" title="c6">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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="c8">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="E1:42_1_2"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">A</font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:42_1_2"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/>
<a NAME="E3:42_1_2"/>
<font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">x</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="E4:42_1_2"/>
<font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 ;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:42"/><i><font color="Green" title="E22">A4</font></i>: 
(  <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="c4">A</font> &amp;  <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="c4">A</font> )
 
<b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/>
<a NAME="E4:42"/>
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> <font color="Maroon" title="c6">C</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</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> <font color="Maroon" title="c6">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">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="E1:42_2"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">x'</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">y'</font> = <font color="Maroon" title="c8">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="c11">x1</font> = <font color="Maroon" title="c9">x'</font>, <font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c10">y'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">A</font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:42_2"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/>
<a NAME="E4:42_2"/>
<font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">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="E5:42_2"/>
<font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:42"/>
( <font color="Maroon" title="c5">C</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">B</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c5">C</font> is <a href="polyalg1.html#V3" title="POLYALG1:attr.3">opers_closed</a> )
 <b>by </b><i><a class="txt" href="polyalg1.html#E2:42"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="polyalg1.html#E3:42"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="polyalg1.html#D4" target="_self" title="POLYALG1:def.4">Def4</a></i>;<br/>


</div>
