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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="Red">::  thesis: </font></span></a><span class="hide"> 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#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> holds <br/> (  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Olive" title="b1">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <span class="p1">(<span class="default"><a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font></span>)</span>,<font color="Olive" title="b3">w</font> &amp;  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">u</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be    <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <span class="p1">(<span class="default"><a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font></span>)</span>,<font color="Maroon" title="c4">w</font> &amp;  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">u</font> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E2:33"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E4:33"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E6:33"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></span>;<br/>
<span class="lab"><font color="Green" title="E27">A4</font></span>:  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> = 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K7" title="QUOFIELD:func.7">pmult</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="quofield.html#T12" target="_self" title="QUOFIELD:th.12">Th12</a></span>
<br/>.= 
 <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="quofield.html#T12" target="_self" title="QUOFIELD:th.12">Th12</a></span>
;<br/>
 <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span> = 
 <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p2">(<span class="default"><a href="quofield.html#K7" title="QUOFIELD:func.7">pmult</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="quofield.html#T12" target="_self" title="QUOFIELD:th.12">Th12</a></span>
<br/>.= 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K7" title="QUOFIELD:func.7">pmult</a> <font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default"><a href="quofield.html#K7" title="QUOFIELD:func.7">pmult</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T12" target="_self" title="QUOFIELD:th.12">Th12</a></span>
<br/>.= 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K7" title="QUOFIELD:func.7">pmult</a> <span class="p2">(<span class="default"><a href="quofield.html#K7" title="QUOFIELD:func.7">pmult</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T5" target="_self" title="QUOFIELD:th.5">Th5</a></span>
<br/>.= 
 <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p2">(<span class="default"><a href="quofield.html#K7" title="QUOFIELD:func.7">pmult</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T12" target="_self" title="QUOFIELD:th.12">Th12</a></span>
<br/>.= 
 <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <span class="p1">(<span class="default"><a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font></span>)</span>,<font color="Maroon" title="c4">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="quofield.html#T12" target="_self" title="QUOFIELD:th.12">Th12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:33"/>
(  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <span class="p1">(<span class="default"><a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font></span>)</span>,<font color="Maroon" title="c4">w</font> &amp;  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E27">A4</font></span></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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