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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="algstr_0.html#V6" title="ALGSTR_0:attr.6">right_add-cancelable</a> <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/>

<b>assume </b><a NAME="E1:2_1_1"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E1:2_1_1_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">y</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="c1">L</font>
 ;<br/><b>thus </b><a NAME="E2:2_1_1_1"/>
( <font color="Maroon" title="c2">x</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="c1">L</font> or <font color="Maroon" title="c3">y</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="c1">L</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:2_1_1_1_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c2">x</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="c1">L</font>
 <b>by </b><i><a class="txt" href="polynom4.html#E1:2_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="vectsp_1.html#D20" title="VECTSP_1:def.20">VECTSP_1:def 20</a></i>;<br/>
<font color="Maroon" title="c4">z</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="polynom4.html#E1:2_1_1_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="polynom4.html#E3:2_1_1_1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>
;<br/>
<b>hence </b><a NAME="E5:2_1_1_1_1"/>
<font color="Maroon" title="c3">y</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="c1">L</font>
 <b>by </b><i><a class="txt" href="polynom4.html#E4"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a>
 <b>by </b><i><a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/>


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