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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="grcat_1.html#K27" title="GRCAT_1:func.27">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <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/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</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>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</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>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">g</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">X</font> =  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">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="grcat_1.html#K27" title="GRCAT_1:func.27">GroupCat</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"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">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="c3">f</font> implies (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</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">g</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:106"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">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="c3">f</font>
 ; <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</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">g</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">g9</font> = <font color="Maroon" title="c4">g</font> as   <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T47" target="_self" title="GRCAT_1:th.47">Th47</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">f9</font> = <font color="Maroon" title="c3">f</font> as   <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T47" target="_self" title="GRCAT_1:th.47">Th47</a></span>;<br/>
<a NAME="E4:106"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
 <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c5">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c6">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="grcat_1.html#T50" target="_self" title="GRCAT_1:th.50">Th50</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">gf</font> = <font color="Maroon" title="c5">g9</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c6">f9</font> as    <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K18" title="GRCAT_1:func.18">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T45" target="_self" title="GRCAT_1:th.45">Th45</a></span>;<br/>
<a NAME="E6:106"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
<font color="Maroon" title="c7">gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="grcat_1.html#T50" target="_self" title="GRCAT_1:th.50">Th50</a></span>;<br/>
<a NAME="E7:106"/>
(  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g9</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c6">f9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c6">f9</font> &amp;  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g9</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c6">f9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c5">g9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:106"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="grcat_1.html#T31" target="_self" title="GRCAT_1:th.31">Th31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:106"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</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">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="grcat_1.html#T50" target="_self" title="GRCAT_1:th.50">Th50</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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