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<b>let </b><font color="Maroon" title="c1">R</font> be   <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">UN</font> being  <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">G</font>,<font color="Olive" title="b4">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b3">G</font> is    <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Olive" title="b1">UN</font>, <font color="Olive" title="b4">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,1 : verum </span>}</span>  &amp;  <a href="modcat_1.html#R1" title="MODCAT_1:pred.1">GO</a> <font color="Olive" title="b2">x</font>, <a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c1">R</font> )</span><br/><b>let </b><font color="Maroon" title="c2">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">G</font> is    <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c2">UN</font>, <font color="Olive" title="b3">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,1 : verum </span>}</span>  &amp;  <a href="modcat_1.html#R1" title="MODCAT_1:pred.1">GO</a> <font color="Olive" title="b1">x</font>, <a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c1">R</font> )</span><br/>



<b>set </b><font color="Maroon" title="c3">T</font> =  <a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font>;<br/>
<a NAME="E1:15"/><i><font color="Green" title="E10">A1</font></i>: 
 <a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# 1,<a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a> ,<a href="funct_5.html#K5" title="FUNCT_5:func.5">op0</a> ,<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,1</span>)</span> #)
 
<b>by </b><i><a class="ref" href="mod_2.html#D2" title="MOD_2:def.2">MOD_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">x1</font> =  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font></span>)</span>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font></span>)</span>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <span class="p1">(<span class="default"><a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font></span>)</span> #) as    <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c2">UN</font> <b>by </b><i><a class="ref" href="grcat_1.html#T42" title="GRCAT_1:th.42">GRCAT_1:42</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">f1</font> = the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p1">(<span class="default"><a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,1 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E10">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x2</font> = <font color="Maroon" title="c5">f1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,1 <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">G</font> is    <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c2">UN</font>, <font color="Olive" title="b2">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,1 : verum </span>}</span>  &amp;  <a href="modcat_1.html#R1" title="MODCAT_1:pred.1">GO</a> <font color="Maroon" title="c7">x</font>, <a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c1">R</font> )</span><br/>

<b>thus </b><a NAME="E5:15"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">G</font> is    <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of  <a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c2">UN</font>, <font color="Olive" title="b2">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,1 : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="modcat_1.html#R1" title="MODCAT_1:pred.1">GO</a> <font color="Maroon" title="c7">x</font>, <a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c1">R</font></span><br/>

<b>thus </b><a NAME="E6:15"/>
 <a href="modcat_1.html#R1" title="MODCAT_1:pred.1">GO</a> <font color="Maroon" title="c7">x</font>, <a href="mod_2.html#K1" title="MOD_2:func.1">TrivialLMod</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="modcat_1.html#D5" target="_self" title="MODCAT_1:def.5">Def5</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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