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<b>let </b><font color="Maroon" title="c2">F</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">left_unital</a> <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Field-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">F</font>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>assume </b><a NAME="E2:34"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">z</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E4:34"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="vectsp_1.html#E1:34"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="vectsp_1.html#D20" target="_self" title="VECTSP_1:def.20">Def20</a></i>;<br/>
<a NAME="E5:34"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">y</font></span>)</span> &amp; <font color="Maroon" title="c6">x1</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">z</font> )
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E6:34"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font>
 
<b>by </b><i><a class="txt" href="vectsp_1.html#E2:34"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="vectsp_1.html#E4:34"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="vectsp_1.html#D19" target="_self" title="VECTSP_1:def.19">Def19</a></i>;<br/>
<b>hence </b><a NAME="E7:34"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="vectsp_1.html#E4:34"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="vectsp_1.html#D19" target="_self" title="VECTSP_1:def.19">Def19</a></i>;<br/>


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