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<b>let </b><font color="Maroon" title="c12">L</font> be  non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">L</font> is <a href="realset2.html#V1" title="REALSET2:attr.1">Field-like</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> implies ( <font color="Maroon" title="c12">L</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> ) )</span><br/>

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c12">L</font> is <a href="realset2.html#V1" title="REALSET2:attr.1">Field-like</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">L</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> )</span><br/>

<b>thus </b><a NAME="E2:45_1_1"/>
<font color="Maroon" title="c12">L</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">L</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:45_1_1_1"/>
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="c12">L</font> holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="realset2.html#T24" target="_self" title="REALSET2:th.24">Th24</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:45_1_1"/>
<font color="Maroon" title="c12">L</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:45_1_1_2"/>
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="c12">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="realset2.html#T23" target="_self" title="REALSET2:th.23">Th23</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:45_1_1"/>
<font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:45_1_1_3"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">L</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c12">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c12">L</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="realset2.html#T25" target="_self" title="REALSET2:th.25">Th25</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c13">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D39" title="ALGSTR_0:def.39">ALGSTR_0:def 39</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">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="c12">L</font> or <font color="Maroon" title="c13">x</font> is <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a> )</span><br/>

<b>assume </b><a NAME="E5:45_1_1"/>
<font color="Maroon" title="c13">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="c12">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> is <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a></span><br/>

<a NAME="E6:45_1_1"/><b>then </b>
not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span><b> such that </b><br/><a NAME="E9:45_1_1"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">y</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="c12">L</font> &amp; <font color="Maroon" title="c15">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">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="c12">L</font> )
 <b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="realset2.html#T11" target="_self" title="REALSET2:th.11">Th11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">y</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D27" title="ALGSTR_0:def.27">ALGSTR_0:def 27</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">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="c12">L</font></span><br/>

<b>thus </b><a NAME="E10:45_1_1"/>
<font color="Maroon" title="c15">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">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="c12">L</font>
 <b>by </b><i><a class="txt" href="#E9:45_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
