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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">F2</font>, <font color="Olive" title="b3">T</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>  st <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/>( <font color="Olive" title="b3">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> iff  for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">f1</font>, <font color="Olive" title="b6">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b5">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b6">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b5">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b4">f</font> &amp; <font color="Olive" title="b4">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b6">f2</font> holds <br/>( <font color="Olive" title="b3">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b5">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f</font> &amp; <font color="Olive" title="b2">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f2</font> &amp; <font color="Olive" title="b3">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b5">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f</font></span>)</span> &amp; <font color="Olive" title="b3">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b6">f2</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<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">  for <font color="Olive" title="b1">T</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>  st <font color="Maroon" title="c3">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/>( <font color="Olive" title="b1">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> iff  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">f1</font>, <font color="Olive" title="b4">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b3">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b4">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b3">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b4">f2</font> holds <br/>( <font color="Olive" title="b1">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f2</font> &amp; <font color="Olive" title="b1">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f</font></span>)</span> &amp; <font color="Olive" title="b1">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b4">f2</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">T</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<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> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  implies ( <font color="Maroon" title="c5">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> iff  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b2">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b3">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">f2</font> holds <br/>( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font></span>)</span> ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:96"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
( <font color="Maroon" title="c3">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</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"> ( <font color="Maroon" title="c5">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> iff  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b2">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b3">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">f2</font> holds <br/>( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font></span>)</span> ) )</span><br/>

<div><span class="kw">hereby </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"> ( (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b2">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b3">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">f2</font> holds <br/>( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font></span>)</span> ) ) implies <font color="Maroon" title="c5">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:96_1"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c5">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</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">f</font>, <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b2">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b3">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">f2</font> holds <br/>( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">f1</font>, <font color="Maroon" title="c8">f2</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"> ( <font color="Maroon" title="c7">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c8">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c8">f2</font> implies ( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f2</font></span>)</span> ) )</span><br/><span class="kw">assume </span><a NAME="E2:96_1"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
( <font color="Maroon" title="c7">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c8">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</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"> ( <font color="Maroon" title="c7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c8">f2</font> implies ( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f2</font></span>)</span> ) )</span><br/><span class="kw">assume </span><a NAME="E3:96_1"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
( <font color="Maroon" title="c7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c8">f2</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="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f2</font></span>)</span> )</span><br/><span class="kw">hence </span><a NAME="E4:96_1"/>
( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E55">A2</font></span></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"> ( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f2</font></span>)</span> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> = 
<font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E3:96_1"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E55">A2</font></span></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"> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f2</font></span>)</span></span><br/><span class="kw">thus </span><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font> = 
<font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c8">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E3:96_1"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="cat_8.html#T4" target="_self" onmouseover="rs('cat_8/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E55">A2</font></span></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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:96"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b2">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b3">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">f2</font> holds <br/>( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b3">f2</font></span>)</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"> <font color="Maroon" title="c5">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><br/>

<a NAME="E4:96"/>
 for <font color="Olive" title="b1">g1</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="c1">C1</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> holds <br/>( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="96_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</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"> ( <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">g2</font> implies ( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_2"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">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"> ( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> )</span><br/>

<a NAME="E2:96_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A7</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">then </span><span class="kw">consider </span><font color="Maroon" title="c8">f1</font>, <font color="Maroon" title="c9">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font><span class="kw"> such that </span><br/><a NAME="E4:96_2"/><span class="lab"><font color="Green" title="E58">A8</font></span>: 
( <font color="Maroon" title="c8">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c9">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 <span class="kw">and </span><br/><a NAME="E5:96_2"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
( <font color="Maroon" title="c8">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">f2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T5" target="_self" onmouseover="rs('cat_8/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">g11</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font><span class="kw"> such that </span><br/><a NAME="E7:96_2"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g11</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c10">g11</font> &amp; <font color="Maroon" title="c10">g11</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_2"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="cat_6.html#D18" onmouseover="rs('cat_6/D18')" onmouseout="rh()">CAT_6:def 18</a></span>;<br/>
<a NAME="E8:96_2"/>
<font color="Maroon" title="c8">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E5:96_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="cat_8.html#T3" target="_self" onmouseover="rs('cat_8/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E9:96_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A11</font></span>: 
( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_2"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E7:96_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E55">A5</font></span></a></span>;<br/>
<a NAME="E10:96_2"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
<font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E1:96_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="cat_8.html#T13" target="_self" onmouseover="rs('cat_8/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:96_2"/>
<font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_2"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="ref" href="cat_8.html#T3" target="_self" onmouseover="rs('cat_8/T3')" onmouseout="rh()">Th3</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"> ( <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">g22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font><span class="kw"> such that </span><br/><a NAME="E13:96_2"/><span class="lab"><font color="Green" title="E63">A13</font></span>: 
(  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g22</font> &amp; <font color="Maroon" title="c11">g22</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c11">g22</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_2"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="cat_6.html#D19" onmouseover="rs('cat_6/D19')" onmouseout="rh()">CAT_6:def 19</a></span>;<br/>
<a NAME="E14:96_2"/>
<font color="Maroon" title="c7">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c9">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E5:96_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="cat_8.html#T3" target="_self" onmouseover="rs('cat_8/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E15:96_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A14</font></span>: 
( <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f2</font> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:96_2"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="txt" href="#E4:96_2"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E55">A5</font></span></a></span>;<br/>
<a NAME="E16:96_2"/><span class="lab"><font color="Green" title="E65">A15</font></span>: 
<font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E1:96_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="cat_8.html#T13" target="_self" onmouseover="rs('cat_8/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:96_2"/>
<font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:96_2"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="ref" href="cat_8.html#T3" target="_self" onmouseover="rs('cat_8/T3')" onmouseout="rh()">Th3</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"> ( <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> &amp; <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_2"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E5:96_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E55">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f1</font></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"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E1:96_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="cat_8.html#T13" target="_self" onmouseover="rs('cat_8/T13')" onmouseout="rh()">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_2"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E10:96_2"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="cat_8.html#T1" target="_self" onmouseover="rs('cat_8/T1')" onmouseout="rh()">Th1</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"> <font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_2"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E5:96_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E55">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E1:96_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="cat_8.html#T13" target="_self" onmouseover="rs('cat_8/T13')" onmouseout="rh()">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:96_2"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="txt" href="#E16:96_2"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="ref" href="cat_8.html#T1" target="_self" onmouseover="rs('cat_8/T1')" onmouseout="rh()">Th1</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:96"/>
<font color="Maroon" title="c5">T</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</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"> verum</span><br/>


</div>
