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<b>let </b><font color="Maroon" title="c1">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/>

<b>set </b><font color="Maroon" title="c2">C</font> =  <a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font>;<br/>
<b>set </b><font color="Maroon" title="c3">V</font> =  <a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font>;<br/>
<b>set </b><font color="Maroon" title="c4">X</font> =  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c5">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">f'</font> be    <a href="ringcat1.html#M3" title="RINGCAT1:mode.3">Element</a> of  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">b'</font> be    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of  <a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font>;<br/>







<a NAME="E1:71"/>
 ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of  <a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Olive" title="b2">H</font> &amp; <font color="Maroon" title="c5">f</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> )
 
<b>by </b><i><a class="ref" href="ringcat1.html#D18" target="_self" title="RINGCAT1:def.18">Def18</a></i>;<br/>
<b>hence </b><a NAME="E2:71"/>
<font color="Maroon" title="c5">f</font> is  <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a>  <a href="ringcat1.html#M3" title="RINGCAT1:mode.3">Element</a> of  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E3:71"/>
<font color="Maroon" title="c6">f'</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:71"/><i><font color="Green" title="E46">A1</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">UN</font> &amp;  <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="ref" href="ringcat1.html#D17" target="_self" title="RINGCAT1:def.17">Def17</a></i>;<br/>
<a NAME="E6:71"/>
 ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font>, <font color="Olive" title="b4">x4</font>, <font color="Olive" title="b5">x5</font>, <font color="Olive" title="b6">x6</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Olive" title="b5">x5</font>,<font color="Olive" title="b6">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp;  ex <font color="Olive" title="b7">G</font> being <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">G</font> &amp; <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b7">G</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b7">G</font> &amp; <font color="Olive" title="b3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b7">G</font> &amp; <font color="Olive" title="b4">x4</font> <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="Olive" title="b7">G</font> &amp; <font color="Olive" title="b5">x5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b7">G</font> &amp; <font color="Olive" title="b6">x6</font> <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="Olive" title="b7">G</font> ) )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E5:71"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="ringcat1.html#D16" target="_self" title="RINGCAT1:def.16">Def16</a></i>;<br/>
<b>hence </b><a NAME="E7:71"/>
<font color="Maroon" title="c7">b</font> is  <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a>  <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of  <a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font>
 ;<br/>

<b>thus </b><a NAME="E8:71"/>
<font color="Maroon" title="c8">b'</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 ;<br/>


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