<?xml version="1.0"?>
<div class="add">

<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"> 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#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;  <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="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#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;  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> )</span><br/>



<b>consider </b><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><b> such that </b><br/><a NAME="E2:30"/><i><font color="Green" title="E23">A1</font></i>: 
 <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</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>
 <b>by </b><i><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">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><b> such that </b><br/><a NAME="E4:30"/><i><font color="Green" title="E24">A2</font></i>: 
<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="c4">y</font>
 <b>by </b><i><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></i>;<br/>
<a NAME="E5:30"/><i><font color="Green" title="E25">A3</font></i>: 
 <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> <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="c4">y</font>,<font color="Maroon" title="c3">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E4:30"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="quofield.html#T11" target="_self" title="QUOFIELD:th.11">Th11</a></i>;<br/>
<a NAME="E6:30"/>
<font color="Maroon" title="c3">x</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>
 
<b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="quofield.html#T6" target="_self" title="QUOFIELD:th.6">Th6</a></i>;<br/>
<a NAME="E7:30"/><b>then </b><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c3">x</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>
 
<b>by </b><i><a class="ref" href="quofield.html#D8" target="_self" title="QUOFIELD:def.8">Def8</a></i>;<br/>
<a NAME="E8:30"/><i><font color="Green" title="E27">A5</font></i>: 
( <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="c4">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> )
 
<b>by </b><i><a class="ref" href="quofield.html#T2" target="_self" title="QUOFIELD:th.2">Th2</a></i>;<br/>
<a NAME="E9:30"/><b>then </b>
<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="c4">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>
 
<b>by </b><i><a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">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="c4">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="c4">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="c4">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> <b>by </b><i><a class="ref" href="quofield.html#D1" target="_self" title="QUOFIELD:def.1">Def1</a></i>;<br/>
<a NAME="E11:30"/><i><font color="Green" title="E28">A6</font></i>: 
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="c4">y</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="c5">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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="c4">y</font> implies <font color="Maroon" title="c6">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="c5">t</font> )</span><br/>

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon" title="c6">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="c4">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">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="c5">t</font></span><br/>

<a NAME="E2:30_1"/><b>then </b><i><font color="Green" title="E29">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c4">y</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="c6">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="c4">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c5">t</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><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"><font color="Maroon" title="c4">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">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="p2">(<span class="default"><font color="Maroon" title="c4">y</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="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E2:30_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">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="p2">(<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>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">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"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">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="c4">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:30"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c5">t</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:30_1"/>
<font color="Maroon" title="c6">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="c5">t</font>
 <b>by </b><i><a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</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="E12:30"/><i><font color="Green" title="E29">A8</font></i>: 
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="c5">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#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c4">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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="c5">t</font> implies <font color="Maroon" title="c6">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="c4">y</font> )</span><br/>

<b>assume </b><a NAME="E1:30_2"/>
<font color="Maroon" title="c6">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="c5">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">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="c4">y</font></span><br/>

<a NAME="E2:30_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c5">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="c6">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="c5">t</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></i>;<br/>
<a NAME="E3:30_2"/><b>then </b><i><font color="Green" title="E30">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><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"><font color="Maroon" title="c4">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c5">t</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<i><font color="Green" title="E31">A10</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">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="p2">(<span class="default"><font color="Maroon" title="c4">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="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><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"><font color="Maroon" title="c4">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">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"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font></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">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:30"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="#E3:30_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">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"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">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="p2">(<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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">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="p2">(<span class="default"><font color="Maroon" title="c4">y</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="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
;<br/>
<a NAME="E5:30_2"/>
<font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
 
;<br/>
<a NAME="E6:30_2"/><b>then </b><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">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="p2">(<span class="default"><font color="Maroon" title="c4">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="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" title="GCD_1:th.7">GCD_1:7</a></i>;<br/>
<a NAME="E7:30_2"/>
<font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
 
;<br/>
<a NAME="E8:30_2"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">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="p2">(<span class="default"><font color="Maroon" title="c4">y</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="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" title="GCD_1:th.7">GCD_1:7</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c4">y</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"><font color="Maroon" title="c6">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="p2">(<span class="default"><font color="Maroon" title="c4">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="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">I</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">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="p2">(<span class="default"><font color="Maroon" title="c4">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="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="gcd_1.html#K1" title="GCD_1:func.1">/</a> <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></span>)</span>
<b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="gcd_1.html#T9" title="GCD_1:th.9">GCD_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">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="p3">(<span class="default"><font color="Maroon" title="c4">y</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="c3">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.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>
<b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="#E4:30_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="#E6:30_2"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="gcd_1.html#T11" title="GCD_1:th.11">GCD_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">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="p2">(<span class="default"><font color="Maroon" title="c4">y</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="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="gcd_1.html#K1" title="GCD_1:func.1">/</a> <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></span>)</span>
<b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="#E8:30_2"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="gcd_1.html#T11" title="GCD_1:th.11">GCD_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">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="p2">(<span class="default"><font color="Maroon" title="c4">y</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="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">I</font></span>)</span>
<b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="gcd_1.html#T9" title="GCD_1:th.9">GCD_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c4">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E10:30_2"/>
<font color="Maroon" title="c6">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="c4">y</font>
 <b>by </b><i><a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</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="E13:30"/>
 <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> <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="c4">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E4:30"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="quofield.html#T11" target="_self" title="QUOFIELD:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E14:30"/>
(  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp;  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> )
 <b>by </b><i><a class="txt" href="#E4:30"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E5:30"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="#E11:30"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="#E12:30"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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