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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>; <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">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="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</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="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/>

<b>set </b><font color="Maroon" title="c2">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="c3">f</font>, <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">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="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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">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="c4">g</font> &amp;  <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 <font color="Maroon" title="c5">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font> )</span><br/>

<b>assume </b><a NAME="E1:75"/><i><font color="Green" title="E50">A1</font></i>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">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="c4">g</font> &amp;  <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">f'</font> = <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c7">g'</font> = <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c8">h'</font> = <font color="Maroon" title="c5">h</font> as  <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> <b>by </b><i><a class="ref" href="ringcat1.html#T27" target="_self" title="RINGCAT1:th.27">Th27</a></i>;<br/>
<a NAME="E3:75"/><i><font color="Green" title="E51">A2</font></i>: 
(  <a href="ringcat1.html#K11" title="RINGCAT1:func.11">dom</a> <font color="Maroon" title="c8">h'</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="c7">g'</font> &amp;  <a href="ringcat1.html#K11" title="RINGCAT1:func.11">dom</a> <font color="Maroon" title="c7">g'</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">f'</font> )
 
<b>by </b><i><a class="txt" href="#E1:75"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="ringcat1.html#T30" target="_self" title="RINGCAT1:th.30">Th30</a></i>;<br/>
<a NAME="E4:75"/><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c7">g'</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c6">f'</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> &amp; <font color="Maroon" title="c8">h'</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c7">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">g</font> )
 
<b>by </b><i><a class="txt" href="#E1:75"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="ringcat1.html#T30" target="_self" title="RINGCAT1:th.30">Th30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">gf</font> = <font color="Maroon" title="c7">g'</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c6">f'</font>, <font color="Maroon" title="c10">hg</font> = <font color="Maroon" title="c8">h'</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c7">g'</font> as  <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> <b>by </b><i><a class="txt" href="#E3:75"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="ringcat1.html#T25" target="_self" title="RINGCAT1:th.25">Th25</a></i>;<br/>
<a NAME="E6:75"/><i><font color="Green" title="E53">A4</font></i>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">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="c4">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:75"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="ringcat1.html#E63"><i><font color="Green" title="E48">Lm9</font></i></a></i>;<br/>
<a NAME="E7:75"/><i><font color="Green" title="E54">A5</font></i>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">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="c3">f</font>
 
<b>by </b><i><a class="txt" href="#E1:75"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="ringcat1.html#E63"><i><font color="Green" title="E48">Lm9</font></i></a></i>;<br/>
<font color="Maroon" title="c5">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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> = 
<font color="Maroon" title="c8">h'</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c9">gf</font>
<b>by </b><i><a class="txt" href="#E4:75"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="#E6:75"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="ringcat1.html#T30" target="_self" title="RINGCAT1:th.30">Th30</a></i>
<br/>.= 
<font color="Maroon" title="c10">hg</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c6">f'</font>
<b>by </b><i><a class="txt" href="#E3:75"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="ringcat1.html#T13" target="_self" title="RINGCAT1:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="txt" href="#E4:75"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="#E7:75"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="ringcat1.html#T30" target="_self" title="RINGCAT1:th.30">Th30</a></i>
;<br/>
<b>hence </b><a NAME="E9:75"/>
<font color="Maroon" title="c5">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">f</font>
 ; <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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