<?xml version="1.0"?>
<div class="add">

<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> 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="b2">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span>  st <font color="Olive" title="b3">f</font> <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"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp;  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  holds <br/>( <font color="Olive" title="b3">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  iff ( <font color="Olive" title="b1">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b2">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f1</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> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span>  st <font color="Olive" title="b2">f</font> <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"><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> &amp;  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  holds <br/>( <font color="Olive" title="b2">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  iff ( <font color="Maroon" title="c3">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b1">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f2</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">  for <font color="Olive" title="b1">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span>  st <font color="Olive" title="b1">f</font> <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"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp;  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  iff ( <font color="Maroon" title="c3">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</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">f</font> <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"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</a></span> &amp;  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  implies ( <font color="Maroon" title="c5">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  iff ( <font color="Maroon" title="c3">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:93"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c5">f</font> <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"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font></span><a href="cat_8.html#K12" title="CAT_8:func.12">]</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="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or ( <font color="Maroon" title="c5">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  iff ( <font color="Maroon" title="c3">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:93"/><span class="lab"><font color="Green" title="E53">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>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  iff ( <font color="Maroon" title="c3">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  ) )</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"> ( <font color="Maroon" title="c3">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c5">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:93_1"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c5">f</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="c3">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c4">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/><a NAME="E2:93_1"/>
( <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E52">A1</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="E3:93_1"/>
( <font color="Maroon" title="c3">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c4">f2</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="#E1:93_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</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="E4:93"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
( <font color="Maroon" title="c3">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c4">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="c5">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/>

<a NAME="E5:93"/>
 for <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span>  st <font color="Maroon" title="c5">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="93_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">g</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</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">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">g</font> implies <font color="Maroon" title="c5">f</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:93_2"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">g</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">f</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c8">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E3:93_2"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
<font color="Maroon" title="c6">g</font> <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"><font color="Maroon" title="c7">g1</font>,<font color="Maroon" title="c8">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="ref" href="cat_8.html#T52" target="_self" onmouseover="rs('cat_8/T52')" onmouseout="rh()">Th52</a></span>;<br/>
<a NAME="E4:93_2"/><span class="lab"><font color="Green" title="E57">A7</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="c7">g1</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c8">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E1:93_2"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E52">A1</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="kw">hence </span><font color="Maroon" title="c5">f</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c6">g</font> = 
<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="c7">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c8">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="#E3:93_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="cat_8.html#T55" target="_self" onmouseover="rs('cat_8/T55')" onmouseout="rh()">Th55</a></span>
<br/>.= 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c7">g1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c8">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="#E4:93_2"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="txt" href="#E4:93"><span class="lab"><font color="Green" title="E54">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/>.= 
<font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E4:93_2"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="txt" href="#E4:93"><span class="lab"><font color="Green" title="E54">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/> <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>
<a NAME="E6:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A8</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a></span>;<br/>
<a NAME="E7:93"/>
 for <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span>  st <font color="Olive" title="b1">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">f</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="93_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">g</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</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="c6">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:93_3"/><span class="lab"><font color="Green" title="E56">A9</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">f</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">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c8">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E3:93_3"/><span class="lab"><font color="Green" title="E57">A10</font></span>: 
<font color="Maroon" title="c6">g</font> <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"><font color="Maroon" title="c7">g1</font>,<font color="Maroon" title="c8">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="ref" href="cat_8.html#T52" target="_self" onmouseover="rs('cat_8/T52')" onmouseout="rh()">Th52</a></span>;<br/>
<a NAME="E4:93_3"/><span class="lab"><font color="Green" title="E58">A11</font></span>: 
( <font color="Maroon" title="c7">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">f1</font> &amp; <font color="Maroon" title="c8">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">f2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_3"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E1:93_3"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E52">A1</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="kw">hence </span><font color="Maroon" title="c6">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c5">f</font> = 
<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="c7">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f2</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="#E3:93_3"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="cat_8.html#T55" target="_self" onmouseover="rs('cat_8/T55')" onmouseout="rh()">Th55</a></span>
<br/>.= 
<span class="p1"><a href="cat_8.html#K12" title="CAT_8:func.12">[</a><span class="default"><font color="Maroon" title="c7">g1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">f2</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="#E4:93_3"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="#E4:93"><span class="lab"><font color="Green" title="E54">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/>.= 
<font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_3"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E4:93_3"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="#E4:93"><span class="lab"><font color="Green" title="E54">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/> <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="E8:93"/>
<font color="Maroon" title="c5">f</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="#E6:93"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="ref" href="cat_6.html#D5" onmouseover="rs('cat_6/D5')" onmouseout="rh()">CAT_6:def 5</a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</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>
