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<span class="kw">let </span><font color="Maroon" title="c1">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <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> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</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> iff  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> )</span><br/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</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> iff  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">f9</font> = <font color="Maroon" title="c4">f</font> as    <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/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">g9</font> = <font color="Maroon" title="c5">g</font> as    <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/>
<a NAME="E3:75"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K11" title="RINGCAT1:func.11">dom</a> <font color="Maroon" title="c7">g9</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K12" title="RINGCAT1:func.12">cod</a> <font color="Maroon" title="c6">f9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ringcat1.html#D20" target="_self" title="RINGCAT1:def.20">Def20</a>, <a class="ref" href="ringcat1.html#D21" target="_self" title="RINGCAT1:def.21">Def21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:75"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</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> iff  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ringcat1.html#D23" target="_self" title="RINGCAT1:def.23">Def23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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