<?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>;<br/>

<a NAME="E1:92"/><i><font color="Green" title="E69">A1</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</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#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font>
 
;<br/>
<a NAME="E2:92"/>
 <a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<a NAME="E1:92_1"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font></span>)</span> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:92_1_1"/><i><font color="Green" title="E70">A2</font></i>: 
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font></span>)</span> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c5">x2</font> = <font color="Maroon" title="c3">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">t1</font> =  <a href="quofield.html#K23" title="QUOFIELD:func.23">quotient</a> <font color="Maroon" title="c4">x1</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font></span>)</span>, <font color="Maroon" title="c7">t2</font> =  <a href="quofield.html#K23" title="QUOFIELD:func.23">quotient</a> <font color="Maroon" title="c5">x2</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font></span>)</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> ;<br/>
<i><font color="Green" title="E71">A3</font></i>:  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t1</font> = 
<span class="p1">(<span class="default"><a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x2</font>
<b>by </b><i><a class="txt" href="quofield.html#E1:92_1_1"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="quofield.html#D28" target="_self" title="QUOFIELD:def.28">Def28</a></i>
<br/>.= 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c7">t2</font>
<b>by </b><i><a class="ref" href="quofield.html#D28" target="_self" title="QUOFIELD:def.28">Def28</a></i>
;<br/>
<a NAME="E5:92_1_1"/>
<font color="Maroon" title="c6">t1</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">t1</font>
 
<b>by </b><i><a class="ref" href="quofield.html#T6" target="_self" title="QUOFIELD:th.6">Th6</a></i>;<br/>
<a NAME="E6:92_1_1"/><b>then </b><i><font color="Green" title="E72">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</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="c7">t2</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">t1</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="c7">t2</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="quofield.html#E4:92_1_1"><i><font color="Green" title="E71">A3</font></i></a>, <a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></i>;<br/>
<a NAME="E7:92_1_1"/><i><font color="Green" title="E73">A5</font></i>: 
<font color="Maroon" title="c7">t2</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon" title="c7">t2</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x2</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">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> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
<b>by </b><i><a class="txt" href="quofield.html#E1:92"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="quofield.html#D27" target="_self" title="QUOFIELD:def.27">Def27</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:92_1_1"/><i><font color="Green" title="E74">A6</font></i>: 
<font color="Maroon" title="c6">t1</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon" title="c6">t1</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">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> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
<b>by </b><i><a class="txt" href="quofield.html#E1:92"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="quofield.html#D27" target="_self" title="QUOFIELD:def.27">Def27</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:92_1_1"/><i><font color="Green" title="E75">A7</font></i>: 
<font color="Maroon" title="c7">t2</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="c5">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_4"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon" title="c7">t2</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x2</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">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> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
<b>by </b><i><a class="txt" href="quofield.html#E1:92"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="quofield.html#D27" target="_self" title="QUOFIELD:def.27">Def27</a></i>
<br/>.= 
<font color="Maroon" title="c5">x2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:92_1_1"/>
<font color="Maroon" title="c6">t1</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">x1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_5"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon" title="c6">t1</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">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> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
<b>by </b><i><a class="txt" href="quofield.html#E1:92"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="quofield.html#D27" target="_self" title="QUOFIELD:def.27">Def27</a></i>
<br/>.= 
<font color="Maroon" title="c4">x1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<b>then </b><font color="Maroon" title="c4">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</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="c7">t2</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E6:92_1_1"><i><font color="Green" title="E72">A4</font></i></a>, <a class="txt" href="quofield.html#E7:92_1_1"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
<br/>.= 
<font color="Maroon" title="c5">x2</font>
<b>by </b><i><a class="txt" href="quofield.html#E8:92_1_1"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="quofield.html#E9:92_1_1"><i><font color="Green" title="E75">A7</font></i></a>, <a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
;<br/>
<b>hence </b><a NAME="E12:92_1_1"/>
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:92_1"/><b>then </b><i><font color="Green" title="E70">A8</font></i>: 
 <a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E3:92_1"/>
 <a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 
<b>by </b><i><a class="ref" href="quofield.html#T59" target="_self" title="QUOFIELD:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E4:92_1"/>
 <a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 <b>by </b><i><a class="txt" href="quofield.html#E2:92_1"><i><font color="Green" title="E70">A8</font></i></a>, <a class="ref" href="quofield.html#D23" target="_self" title="QUOFIELD:def.23">Def23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:92"/>
 <a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 ;<br/>


</div>
