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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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 <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp;  <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <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#V13" title="ALGSTR_0:attr.13">right_complementable</a> )</span><br/>

<a NAME="E1:58"/><i><font color="Green" title="E46">A1</font></i>: 
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<a NAME="E1:58_1"/>
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span>
 
<b>by </b><i><a class="ref" href="quofield.html#T22" target="_self" title="QUOFIELD:th.22">Th22</a></i>;<br/>
<b>hence </b><a NAME="E2:58_1"/>
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</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><b>end;</b></div>
<a NAME="E2:58"/><i><font color="Green" title="E47">A2</font></i>: 
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<a NAME="E1:58_2"/>
for <font color="Olive" title="b1">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">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
 
<b>by </b><i><a class="ref" href="quofield.html#T24" target="_self" title="QUOFIELD:th.24">Th24</a></i>;<br/>
<b>hence </b><a NAME="E2:58_2"/>
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</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><b>end;</b></div>
<a NAME="E3:58"/>
 <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#V13" title="ALGSTR_0:attr.13">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">v</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">v</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c3">m</font> = <font color="Maroon" title="c2">v</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K9" title="QUOFIELD:func.9">Quot.</a> <font color="Maroon" title="c1">I</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">n</font> = <span class="p1">(<span class="default"><a href="quofield.html#K19" title="QUOFIELD:func.19">quotaddinv</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">m</font> as   <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> ;<br/>
<b>take </b><font color="Maroon" title="c5">w</font> = <font color="Maroon" title="c4">n</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</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></span><br/>

<b>thus </b><a NAME="E3:58_3"/>
<font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</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>
 <b>by </b><i><a class="ref" href="quofield.html#T30" target="_self" title="QUOFIELD:th.30">Th30</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><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
(  <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp;  <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <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#V13" title="ALGSTR_0:attr.13">right_complementable</a> )
 <b>by </b><i><a class="txt" href="quofield.html#E1:58"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="quofield.html#E2:58"><i><font color="Green" title="E47">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>
