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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">R</font> being  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" title="MODCAT_1:func.12">LModCat</a> <font color="Maroon" title="c1">UN</font>,<font color="Olive" title="b1">R</font></span>)</span>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">h</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="b3">g</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b3">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="b2">f</font> holds <br/><font color="Olive" title="b4">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</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>, <font color="Olive" title="b3">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" title="MODCAT_1:func.12">LModCat</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b3">h</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="b2">g</font> &amp;  <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> holds <br/><font color="Olive" title="b3">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">X</font> =  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" title="MODCAT_1:func.3">LModObjects</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font>, <font color="Maroon" title="c6">h</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" title="MODCAT_1:func.12">LModCat</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">h</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="c5">g</font> &amp;  <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> implies <font color="Maroon" title="c6">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">h</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="c5">g</font> &amp;  <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">f9</font> = <font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c8">g9</font> = <font color="Maroon" title="c5">g</font>, <font color="Maroon" title="c9">h9</font> = <font color="Maroon" title="c6">h</font> as   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" title="MODCAT_1:func.3">LModObjects</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#T15" target="_self" title="MODCAT_1:th.15">Th15</a></span>;<br/>
<a NAME="E3:40"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
( <font color="Maroon" title="c9">h9</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c8">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">g</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <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="txt" href="#E1:40"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="modcat_1.html#E34"><span class="lab"><font color="Green" title="E27">Lm1</font></span></a>, <a class="ref" href="modcat_1.html#T18" target="_self" title="MODCAT_1:th.18">Th18</a></span>;<br/>
<a NAME="E4:40"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
(  <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c9">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c8">g9</font> &amp;  <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c8">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c7">f9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="modcat_1.html#T18" target="_self" title="MODCAT_1:th.18">Th18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">gf</font> = <font color="Maroon" title="c8">g9</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c7">f9</font>, <font color="Maroon" title="c11">hg</font> = <font color="Maroon" title="c9">h9</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c8">g9</font> as   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" title="MODCAT_1:func.3">LModObjects</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#T12" target="_self" title="MODCAT_1:th.12">Th12</a></span>;<br/>
<a NAME="E6:40"/>
( <font color="Maroon" title="c8">g9</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c7">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="modcat_1.html#E34"><span class="lab"><font color="Green" title="E27">Lm1</font></span></a>, <a class="ref" href="modcat_1.html#T18" target="_self" title="MODCAT_1:th.18">Th18</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c6">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> = 
<font color="Maroon" title="c9">h9</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c10">gf</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#T18" target="_self" title="MODCAT_1:th.18">Th18</a></span>
<br/>.= 
<font color="Maroon" title="c11">hg</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c7">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="mod_2.html#T25" title="MOD_2:th.25">MOD_2:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="modcat_1.html#T18" target="_self" title="MODCAT_1:th.18">Th18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:40"/>
<font color="Maroon" title="c6">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">f</font>
 ; <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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