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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a>  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</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> </span><br/>

<a NAME="E1:66"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is  <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> or <font color="Maroon" title="c2">x</font> is  <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:66_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> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> is  <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:66_1"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T45" target="_self" title="QUOFIELD:th.45">Th48</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">y</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E4:66_1"/>
<font color="Maroon" title="c3">y</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66_1"><span class="lab"><font color="Green" title="E51">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:66"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
(  <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T19" target="_self" title="QUOFIELD:th.19">Th21</a></span>;<br/>
<a NAME="E3:66"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
(  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</a> <font color="Maroon" title="c1">I</font> &amp; (  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 <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><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></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">z</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T26" target="_self" title="QUOFIELD:th.26">Th28</a>, <a class="ref" href="quofield.html#T27" target="_self" title="QUOFIELD:th.27">Th29</a></span>;<br/>
<a NAME="E4:66"/>
( (  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 <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> holds  <span class="p1">(<span class="default"><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></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">z</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> holds  <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">u</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T21" target="_self" title="QUOFIELD:th.21">Th23</a>, <a class="ref" href="quofield.html#T23" target="_self" title="QUOFIELD:th.23">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:66"/>
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a>, <a class="ref" href="rlvect_1.html#D2" title="RLVECT_1:def.2">RLVECT_1:def 2</a>, <a class="ref" href="struct_0.html#D8" title="STRUCT_0:def.8">STRUCT_0:def 8</a>, <a class="ref" href="vectsp_1.html#D7" title="VECTSP_1:def.7">VECTSP_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
