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

<a NAME="E1:26_1_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">I</font> <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>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">I</font></span>)</span>,<span class="p2">(<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></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/>
<b>set </b><font color="Maroon" title="c3">M</font> =  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c2">t</font>;<br/>
<a NAME="E3:26_1_1"/><i><font color="Green" title="E18">A1</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="c2">t</font> holds <br/><font color="Olive" title="b1">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="c4">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="c2">t</font> implies <font color="Maroon" title="c4">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  )</span><br/>

<b>assume </b><a NAME="E1:26_1_1_1"/>
<font color="Maroon" title="c4">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="c2">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="c4">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span><br/>

<a NAME="E2:26_1_1_1"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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/>
<font color="Maroon" title="c4">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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"><font color="Maroon" title="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E2:26_1_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <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"><font color="Maroon" title="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
<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="E4:26_1_1_1"/>
<font color="Maroon" title="c4">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
 ; <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="E4:26_1_1"/>
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="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  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="c2">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="c4">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  implies <font color="Maroon" title="c4">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="c2">t</font> )</span><br/>

<b>assume </b><a NAME="E1:26_1_1_2"/>
<font color="Maroon" title="c4">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">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="c2">t</font></span><br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="E3:26_1_1_2"/>
<font color="Maroon" title="c4">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="c2">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>
<b>hence </b><a NAME="E5:26_1_1"/>
 ex <font color="Olive">b<sub>1</sub></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> st <br/>for <font color="Olive" title="b2">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> holds <br/> ( <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Olive" title="b2">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  )
 <b>by </b><i><a class="txt" href="quofield.html#E3:26_1_1"><i><font color="Green" title="E18">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
