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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">  <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> </span><br/>

<a NAME="E1:91"/><span class="lab"><font color="Green" title="E68">A1</font></span>: 
 <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="E2:91"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <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> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="Olive" title="b1">x</font></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="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="Olive" title="b2">y</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_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="Olive" title="b1">x</font></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"><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="Olive" title="b2">y</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_2.html#K3" title="FUNCT_2:func.3">.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">res3</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font></span>)</span>,<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> 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="txt" href="#E1:91"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="quofield.html#D1" target="_self" title="QUOFIELD:def.1">Def1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="quofield.html#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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="c3">x</font></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="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="c4">y</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_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="c3">x</font></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"><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="c4">y</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_2.html#K3" title="FUNCT_2:func.3">.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">t1</font> =  <a href="quofield.html#K23" title="QUOFIELD:func.23">quotient</a> (<font color="Maroon" title="c3">x</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="c6">t2</font> =  <a href="quofield.html#K23" title="QUOFIELD:func.23">quotient</a> (<font color="Maroon" title="c4">y</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/>
<span class="lab"><font color="Green" title="E69">A2</font></span>: <font color="Maroon" title="c5">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="c3">x</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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="quofield.html#D24" target="_self" title="QUOFIELD:def.24">Def27</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
;<br/>
<a NAME="E4:91_1"/>
( <font color="Maroon" title="c5">t1</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="c6">t2</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="E5:91_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">t1</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">t2</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="c7">sum</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">t1</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="c6">t2</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="c6">t2</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">t1</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="c5">t1</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="c6">t2</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/>
<span class="lab"><font color="Green" title="E71">A4</font></span>: <font color="Maroon" title="c6">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="c4">y</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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="quofield.html#D24" target="_self" title="QUOFIELD:def.24">Def27</a></span>
<br/>.= 
<font color="Maroon" title="c4">y</font>
<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="kw">reconsider </span><font color="Maroon" title="c8">prod</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"><font color="Maroon" title="c5">t1</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="c6">t2</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">t1</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="c6">t2</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="txt" href="#E5:91_1"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="ref" href="quofield.html#D1" target="_self" title="QUOFIELD:def.1">Def1</a></span>;<br/>
<a NAME="E9:91_1"/><span class="lab"><font color="Green" title="E72">A5</font></span>: 
(  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c5">t1</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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> &amp;  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t2</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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D25" target="_self" title="QUOFIELD:def.25">Def28</a></span>;<br/>
<span class="lab"><font color="Green" title="E73">A6</font></span>: <span class="p1">(<span class="default"><a href="quofield.html#K17" title="QUOFIELD:func.17">quotadd</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c5">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t2</font></span>)</span>) = 
 <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c5">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D12" target="_self" title="QUOFIELD:def.12">Def12</a></span>
<br/>.= 
 <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">t1</font>,<font color="Maroon" title="c6">t2</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T9" target="_self" title="QUOFIELD:th.9">Th11</a></span>
<br/>.= 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c7">sum</font>

;<br/>
<span class="lab"><font color="Green" title="E74">A7</font></span>: <font color="Maroon" title="c5">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="c3">x</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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="quofield.html#D24" target="_self" title="QUOFIELD:def.24">Def27</a></span>
<br/>.= 
<font color="Maroon" title="c3">x</font>
<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="lab"><font color="Green" title="E75">A8</font></span>: <font color="Maroon" title="c6">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="c4">y</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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="quofield.html#D24" target="_self" title="QUOFIELD:def.24">Def27</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E76">A9</font></span>: <font color="Maroon" title="c7">sum</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"><font color="Maroon" title="c5">t1</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></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="c6">t2</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"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<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>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="vectsp_1.html#D6" title="VECTSP_1:def.6">VECTSP_1:def 6</a></span>
<br/>.= 
<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"><font color="Maroon" title="c5">t1</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">t2</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<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>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D6" title="VECTSP_1:def.6">VECTSP_1:def 6</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font></span>)</span>,<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91_1"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="txt" href="#E11:91_1"><span class="lab"><font color="Green" title="E74">A7</font></span></a>, <a class="ref" href="vectsp_1.html#D6" title="VECTSP_1:def.6">VECTSP_1:def 6</a></span>
;<br/>
<span class="kw">thus </span><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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font></span>)</span> = 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K23" title="QUOFIELD:func.23">quotient</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font></span>)</span>,<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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D25" target="_self" title="QUOFIELD:def.25">Def28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c3">x</font></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="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="c4">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="txt" href="#E9:91_1"><span class="lab"><font color="Green" title="E72">A5</font></span></a>, <a class="txt" href="#E10:91_1"><span class="lab"><font color="Green" title="E73">A6</font></span></a>, <a class="txt" href="#E13:91_1"><span class="lab"><font color="Green" title="E76">A9</font></span></a>, <a class="ref" href="quofield.html#D24" target="_self" title="QUOFIELD:def.24">Def27</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="quofield.html#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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="c3">x</font></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"><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="c4">y</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_2.html#K3" title="FUNCT_2:func.3">.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> )</span><br/>

<span class="lab"><font color="Green" title="E77">A10</font></span>: <span class="p1">(<span class="default"><a href="quofield.html#K18" title="QUOFIELD:func.18">quotmult</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c5">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t2</font></span>)</span>) = 
 <a href="quofield.html#K11" title="QUOFIELD:func.11">qmult</a> (<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c5">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">t2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D13" target="_self" title="QUOFIELD:def.13">Def13</a></span>
<br/>.= 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K7" title="QUOFIELD:func.7">pmult</a> (<font color="Maroon" title="c5">t1</font>,<font color="Maroon" title="c6">t2</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T10" target="_self" title="QUOFIELD:th.10">Th12</a></span>
<br/>.= 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c8">prod</font>

;<br/>
<a NAME="E16:91_1"/><span class="lab"><font color="Green" title="E78">A11</font></span>: 
<font color="Maroon" title="c8">prod</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span>,<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:91_1"><span class="lab"><font color="Green" title="E75">A8</font></span></a>, <a class="txt" href="#E3:91_1"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="txt" href="#E7:91_1"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="txt" href="#E11:91_1"><span class="lab"><font color="Green" title="E74">A7</font></span></a>, <a class="ref" href="vectsp_1.html#D6" title="VECTSP_1:def.6">VECTSP_1:def 6</a></span>;<br/>
<span class="kw">thus </span><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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span> = 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K23" title="QUOFIELD:func.23">quotient</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span>,<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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D25" target="_self" title="QUOFIELD:def.25">Def28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c3">x</font></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"><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="c4">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="txt" href="#E9:91_1"><span class="lab"><font color="Green" title="E72">A5</font></span></a>, <a class="txt" href="#E15:91_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="txt" href="#E16:91_1"><span class="lab"><font color="Green" title="E78">A11</font></span></a>, <a class="ref" href="quofield.html#D24" target="_self" title="QUOFIELD:def.24">Def27</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="quofield.html#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> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span></span><br/>

<a NAME="E18:91_1"/><span class="lab"><font color="Green" title="E79">A12</font></span>: 
 for <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</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">res3</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</a> <font color="Maroon" title="c1">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c9">u</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">res3</font> implies <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</a> <font color="Maroon" title="c1">I</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_1_10"/><span class="lab"><font color="Green" title="E80">A13</font></span>: 
<font color="Maroon" title="c9">u</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">res3</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="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</a> <font color="Maroon" title="c1">I</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c9">u</font> 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/>
<font color="Maroon" title="c10">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">u</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"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D6" title="VECTSP_1:def.6">VECTSP_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">u</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="c2">res3</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#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">u</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="c2">res3</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_10"><span class="lab"><font color="Green" title="E80">A13</font></span></a>, <a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">u</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font></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/>.= 
<font color="Maroon" title="c10">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D6" title="VECTSP_1:def.6">VECTSP_1:def 6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:91_1_10"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</a> <font color="Maroon" title="c1">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D9" target="_self" title="QUOFIELD:def.9">Def9</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="E19:91_1"/>
 for <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b1">u</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">res3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c9">u</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">res3</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_1_11"/><span class="lab"><font color="Green" title="E80">A14</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</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="c9">u</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">res3</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c9">u</font> 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/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">u</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="c2">res3</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">u</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"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c10">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D6" title="VECTSP_1:def.6">VECTSP_1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c10">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_11"><span class="lab"><font color="Green" title="E80">A14</font></span></a>, <a class="ref" href="quofield.html#D9" target="_self" title="QUOFIELD:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">u</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D6" title="VECTSP_1:def.6">VECTSP_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">u</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="c2">res3</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="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:91_1_11"/>
<font color="Maroon" title="c9">u</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">res3</font>
 <span class="kw">by</span> <span class="lab"><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="E20:91_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A15</font></span>: 
 <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</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="c2">res3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:91_1"><span class="lab"><font color="Green" title="E79">A12</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<span class="kw">thus </span><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> <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> = 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K23" title="QUOFIELD:func.23">quotient</a> (<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 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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D25" target="_self" title="QUOFIELD:def.25">Def28</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="txt" href="#E20:91_1"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="ref" href="quofield.html#D24" target="_self" title="QUOFIELD:def.24">Def27</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="E3:91"/><span class="kw">then </span>
(  <a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  &amp;  <a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp;  <a href="quofield.html#K24" title="QUOFIELD:func.24">canHom</a> <font color="Maroon" title="c1">I</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D8" title="GRCAT_1:def.8">GRCAT_1:def 8</a>, <a class="ref" href="group_1.html#D13" title="GROUP_1:def.13">GROUP_1:def 13</a>, <a class="ref" href="group_6.html#D6" title="GROUP_6:def.6">GROUP_6:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:91"/>
 <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> 
 ; <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>
