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<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">d</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_5.html#K12" title="CAT_5:func.12">Hom</a> <font color="Maroon" title="c2">o</font>; <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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Olive" title="b1">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Olive" title="b2">g</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>,<font color="Olive" title="b3">h</font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">h</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font>),<font color="Olive" title="b1">f</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c11">h</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">g</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c11">h</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">h</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c10">g</font>),<font color="Maroon" title="c9">f</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53_3"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">b</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="c9">f</font>
 <span class="kw">and </span><br/><a NAME="E2:53_3"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">and </span><br/><a NAME="E3:53_3"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">c</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="c10">g</font>
 <span class="kw">and </span><br/><a NAME="E4:53_3"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">and </span><br/><a NAME="E5:53_3"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c8">d</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="c11">h</font>
 <span class="kw">and </span><br/><a NAME="E6:53_3"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c11">h</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">H<sub>1</sub></font>(<font color="Maroon" title="c11">h</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c10">g</font>),<font color="Maroon" title="c9">f</font>)</span><br/>

<a NAME="E7:53_3"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">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="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_3"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E3:53_3"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E4:53_3"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></span>;<br/>
<a NAME="E8:53_3"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">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="c10">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53_3"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E5:53_3"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E6:53_3"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:53_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">h</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c10">g</font>),<font color="Maroon" title="c9">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53_3"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</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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