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<b>set </b><font color="Maroon">M</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E1:8_3_1"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<a NAME="E4:8_3_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</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> <font color="Maroon">M</font>
 
;<br/>
<a NAME="E5:8_3_1"/>
for <font color="Olive">p</font>, <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">p</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
;<br/>
<b>hence </b><a NAME="E6:8_3_1"/>
( <font color="Maroon">M</font> is <a href="ideal_1.html#V1" target="_self">add-closed</a> &amp; <font color="Maroon">M</font> is <a href="ideal_1.html#V2" target="_self">left-ideal</a> )
 <b>by </b><i>, , </i>;<br/>


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