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<span class="kw">let </span><font color="Maroon" title="c2">F</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</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#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">F</font>  st <font color="Olive" title="b1">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="c2">F</font> &amp; <font color="Olive" title="b1">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">z</font> holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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="c2">F</font> &amp; <font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">z</font> implies <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:64"/>
<font color="Maroon" title="c3">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="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">z</font> or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:64"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">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="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D20" target="_self" title="VECTSP_1:def.20">Def20</a></span>;<br/>
<a NAME="E4:64"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span> &amp; <font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:64"/>
<font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font></span><br/>

<a NAME="E6:64"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="vectsp_1.html#D19" target="_self" title="VECTSP_1:def.19">Def19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:64"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="vectsp_1.html#D19" target="_self" title="VECTSP_1:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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