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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">  for <font color="Olive" title="b1">u</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/> ( <span class="p1">(<span class="default"><a href="quofield.html#K17" title="QUOFIELD:func.17">quotadd</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; <span class="p1">(<span class="default"><a href="quofield.html#K17" title="QUOFIELD:func.17">quotadd</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span>)</span>,<font color="Olive" title="b1">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="quofield.html#K17" title="QUOFIELD:func.17">quotadd</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <span class="p1">(<span class="default"><a href="quofield.html#K17" title="QUOFIELD:func.17">quotadd</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span>)</span>,<font color="Maroon" title="c2">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> )</span><br/>



<span class="lab"><font color="Green" title="E37">A1</font></span>: <span class="p1">(<span class="default"><a href="quofield.html#K17" title="QUOFIELD:func.17">quotadd</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span>)</span>,<font color="Maroon" title="c2">u</font>) = 
 <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span>)</span>,<font color="Maroon" title="c2">u</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D12" target="_self" title="QUOFIELD:def.12">Def12</a></span>
<br/>.= 
<font color="Maroon" title="c2">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T12" target="_self" title="QUOFIELD:th.12">Th14</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="quofield.html#K17" title="QUOFIELD:func.17">quotadd</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span>)</span>) = 
 <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> (<font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D12" target="_self" title="QUOFIELD:def.12">Def12</a></span>
<br/>.= 
<font color="Maroon" title="c2">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T12" target="_self" title="QUOFIELD:th.12">Th14</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:44"/>
( <span class="p1">(<span class="default"><a href="quofield.html#K17" title="QUOFIELD:func.17">quotadd</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <span class="p1">(<span class="default"><a href="quofield.html#K17" title="QUOFIELD:func.17">quotadd</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span>)</span>,<font color="Maroon" title="c2">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E37">A1</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/>


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