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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/>

<a NAME="E1:21"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</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="c1">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">cL</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:21"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">cL</font> is <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">IDEAL_1:def 1</a><br/>

<b>thus </b><a NAME="E1:21_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">cL</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">cL</font> implies <font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">cL</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:21"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c2">cL</font> is <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">IDEAL_1:def 2</a><br/>

<b>thus </b><a NAME="E1:21_2"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">cL</font> implies <font color="Maroon" title="c3">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">cL</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:21"/>
<font color="Maroon" title="c2">cL</font> is <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D3" target="_self" title="IDEAL_1:def.3">IDEAL_1:def 3</a><br/>

<b>thus </b><a NAME="E1:21_3"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">cL</font> implies <font color="Maroon" title="c4">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">cL</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:21"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="ideal_1.html#E4:21"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>


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