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<b>set </b><font color="Maroon" title="c2">M</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:8_1_1"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">M</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> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">M</font> = <font color="Maroon" title="c3">M</font> as  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="c1">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">M</font>
;<br/>

<a NAME="E4:8_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
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="c1">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> holds <br/><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 
;<br/>
<a NAME="E5:8_1_1"/><i><font color="Green" title="E6">A2</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> holds <br/><font color="Olive" title="b1">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 
;<br/>
<a NAME="E6:8_1_1"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> holds <br/><font color="Olive" title="b2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 
;<br/>
<b>hence </b><a NAME="E7:8_1_1"/>
( <font color="Maroon" title="c4">M</font> is <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> &amp; <font color="Maroon" title="c4">M</font> is <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> &amp; <font color="Maroon" title="c4">M</font> is <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:8_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="ideal_1.html#E5:8_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">Def1</a>, <a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">Def2</a>, <a class="ref" href="ideal_1.html#D3" target="_self" title="IDEAL_1:def.3">Def3</a></i>;<br/>


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