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<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="struct_0.html#V3">empty</a>  <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font>;<br/>





<b>take </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>;<br/>

<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/>
<a NAME="E2:43"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">C</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_1"/>
for <font color="Olive">v</font>, <font color="Olive">u</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> <font color="Maroon">C</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</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="43_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">u</font>, <font color="Maroon">y</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E3:43_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:43_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:43_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> <font color="Maroon">C</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> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_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="E1:43_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E3:43_1_2"/>
<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">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:43_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> <font color="Maroon">C</font>
 ;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:43"/><i><font color="Green">E47</font></i>: 
(  <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">A</font> &amp;  <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">A</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:43"/>
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> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</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> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_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="E1:43_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<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">A</font> <b>by </b><i/>;<br/>
<a NAME="E4:43_2"/>
<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="E5:43_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> <font color="Maroon">C</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:43"/>
( <font color="Maroon">C</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">B</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> is <a href="polyalg1.html#V3" target="_self">opers_closed</a> )
 <b>by </b><i>, , </i>;<br/>


</div>
