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<span class="kw">let </span><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font><br/>  for <font color="Olive" title="b3">f2</font>, <font color="Olive" title="b4">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font>  st <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Olive" title="b3">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b4">g2</font> holds <br/><span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b3">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Olive" title="b2">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b4">g2</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f2</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font>  st <font color="Maroon" title="c3">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Olive" title="b1">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> holds <br/><span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Olive" title="b2">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f2</font>, <font color="Maroon" title="c6">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">g2</font> implies <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">g2</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:92"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c3">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">g2</font> or <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">g2</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> )</span><br/>

<a NAME="E2:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A2</font></span>: 
 not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:92"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c5">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">g2</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span><br/>

<a NAME="E4:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A4</font></span>: 
 not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/>
<a NAME="E5:92"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="cat_8.html#T54" target="_self" onmouseover="rs('cat_8/T54')" onmouseout="rh()">Th54</a></span>;<br/>
<span class="lab"><font color="Green" title="E56">A6</font></span>: <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:92"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="cat_8.html#T13" target="_self" onmouseover="rs('cat_8/T13')" onmouseout="rh()">Th13</a></span>
<br/>.= 
<font color="Maroon" title="c3">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>
<br/>.= 
<font color="Maroon" title="c3">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:92"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="cat_8.html#T13" target="_self" onmouseover="rs('cat_8/T13')" onmouseout="rh()">Th13</a></span>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">g2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:92"/>
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">g2</font></span>)</span></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E6:92"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="cat_8.html#D23" target="_self" onmouseover="rs('cat_8/D23')" onmouseout="rh()">Def23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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