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<div class="add">

<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/> (  <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> (<font color="Olive" title="b1">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K15" title="QUOFIELD:func.15">qaddinv</a> <font color="Olive" title="b1">u</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> &amp;  <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K15" title="QUOFIELD:func.15">qaddinv</a> <font color="Olive" title="b1">u</font></span>)</span>,<font color="Olive" title="b1">u</font>) <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><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"> (  <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#K15" title="QUOFIELD:func.15">qaddinv</a> <font color="Maroon" title="c2">u</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> &amp;  <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K15" title="QUOFIELD:func.15">qaddinv</a> <font color="Maroon" title="c2">u</font></span>)</span>,<font color="Maroon" title="c2">u</font>) <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><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">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="E2:35"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 <a href="quofield.html#K15" title="QUOFIELD:func.15">qaddinv</a> <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="c3">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/>
<a NAME="E3:35"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K15" title="QUOFIELD:func.15">qaddinv</a> <font color="Maroon" title="c2">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="quofield.html#T5" target="_self" title="QUOFIELD:th.5">Th6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</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="E5:35"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font>
 <span class="kw">and </span><br/><a NAME="E6:35"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D10" target="_self" title="QUOFIELD:def.10">Def10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">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="E8:35"/><span class="lab"><font color="Green" title="E31">A4</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">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/>
<a NAME="E9:35"/>
( <font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <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="c1">I</font> &amp; <font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <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="c1">I</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T2" target="_self" title="QUOFIELD:th.2">Th2</a></span>;<br/>
<a NAME="E10:35"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <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="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_2.html#D1" title="VECTSP_2:def.1">VECTSP_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">t</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <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">by</span> <span class="lab"><a class="ref" href="quofield.html#D1" target="_self" title="QUOFIELD:def.1">Def1</a></span>;<br/>
<a NAME="E12:35"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <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="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T2" target="_self" title="QUOFIELD:th.2">Th2</a></span>;<br/>
<a NAME="E13:35"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="quofield.html#T5" target="_self" title="QUOFIELD:th.5">Th6</a></span>;<br/>
<a NAME="E14:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="quofield.html#T7" target="_self" title="QUOFIELD:th.7">Th8</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D3" title="VECTSP_1:def.3">VECTSP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T48" title="GCD_1:th.48">GCD_1:48</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T48" title="GCD_1:th.48">GCD_1:48</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a></span>
;<br/>
<a NAME="E16:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <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="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="vectsp_2.html#D1" title="VECTSP_2:def.1">VECTSP_2:def 1</a></span>;<br/>
<a NAME="E17:35"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
 for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <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>; <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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</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"> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t</font></span><br/>

<a NAME="E2:35_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <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="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D8" target="_self" title="QUOFIELD:def.8">Def8</a></span>;<br/>
<a NAME="E3:35_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <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="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T7" title="VECTSP_1:th.7">VECTSP_1:7</a></span>;<br/>
<a NAME="E4:35_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <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="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="vectsp_1.html#T7" title="VECTSP_1:th.7">VECTSP_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:35_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_2"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</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="E18:35"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <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="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T2" target="_self" title="QUOFIELD:th.2">Th2</a></span>;<br/>
<a NAME="E19:35"/><span class="lab"><font color="Green" title="E37">A11</font></span>: 
 for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t</font> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <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>; <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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t</font> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_3"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t</font>
 ; <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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font></span><br/>

<a NAME="E2:35_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></span>;<br/>
<a NAME="E3:35_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <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="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="vectsp_1.html#T7" title="VECTSP_1:th.7">VECTSP_1:7</a></span>;<br/>
<a NAME="E4:35_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <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="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E2:35_3"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="ref" href="vectsp_2.html#D1" title="VECTSP_2:def.1">VECTSP_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:35_3"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#D8" target="_self" title="QUOFIELD:def.8">Def8</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="E20:35"/>
(  <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#K15" title="QUOFIELD:func.15">qaddinv</a> <font color="Maroon" title="c2">u</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K6" title="QUOFIELD:func.6">padd</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> &amp;  <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K15" title="QUOFIELD:func.15">qaddinv</a> <font color="Maroon" title="c2">u</font></span>)</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> <span class="p1">(<span class="default"><a href="quofield.html#K6" title="QUOFIELD:func.6">padd</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">y</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="quofield.html#T9" target="_self" title="QUOFIELD:th.9">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:35"/>
(  <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#K15" title="QUOFIELD:func.15">qaddinv</a> <font color="Maroon" title="c2">u</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> &amp;  <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K15" title="QUOFIELD:func.15">qaddinv</a> <font color="Maroon" title="c2">u</font></span>)</span>,<font color="Maroon" title="c2">u</font>) <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="txt" href="#E19:35"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="txt" href="#E17:35"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="subset_1.html#T3" title="SUBSET_1:th.3">SUBSET_1:3</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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