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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:48"/><i><font color="Green">E33</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E2:48"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="cat_1.html#V3" target="_self">monic</a> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D10" target="_self">CAT_1:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E2:48_1"/>
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:48_1"/><b>then </b>
(  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="cat_1.html#D13" target="_self">Def13</a>, <a class="ref" href="cat_1.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E4:48_1"/>
(  not <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="cat_1.html#E1:48_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:48"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D10" target="_self">CAT_1:def 10</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E4:48"/><i><font color="Green">E35</font></i>: 
 <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:48"/><i><font color="Green">E36</font></i>: 
 <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:48"/><i><font color="Green">E37</font></i>: 
 <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E7:48"/><i><font color="Green">E38</font></i>: 
 <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="cat_1.html#T23" target="_self">Th23</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="cat_1.html#M1" target="_self">Morphism</a> of  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_1.html#E5:48"><i><font color="Green">E36</font></i></a>, <a class="txt" href="cat_1.html#E7:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cat_1.html#T22" target="_self">Th22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="cat_1.html#M1" target="_self">Morphism</a> of  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_1.html#E4:48"><i><font color="Green">E35</font></i></a>, <a class="txt" href="cat_1.html#E6:48"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cat_1.html#E7:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cat_1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E10:48"/><i><font color="Green">E39</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_1.html#E5:48"><i><font color="Green">E36</font></i></a>, <a class="txt" href="cat_1.html#E7:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cat_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E11:48"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="cat_1.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E12:48"/>
(  not <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="cat_1.html#E3:48"><i><font color="Green">E34</font></i></a>, <a class="txt" href="cat_1.html#E10:48"><i><font color="Green">E39</font></i></a></i>;<br/>


</div>
