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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</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="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>











<b>assume </b><a NAME="E1:42"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>thus </b><a NAME="E2:42"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="altcat_3.html#V4">mono</a> implies <font color="Maroon">c<sub>8</sub></font> is <a href="altcat_3.html#V4">mono</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="altcat_3.html#V4">mono</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D7">ALTCAT_3:def 7</a><br/>

<b>assume </b><a NAME="E2:42_1"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>

<b>assume </b><a NAME="E3:42_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="altcat_2.html#T30">ALTCAT_2:30</a></i>;<br/>
<a NAME="E5:42_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E6:42_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <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="altcat_4.html#E2:42_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="altcat_4.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_4.html#E2:42_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="altcat_2.html#T34">ALTCAT_2:34</a></i>;<br/>
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="altcat_4.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_4.html#E2:42_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="altcat_4.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_4.html#E2:42_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_4.html#E3:42_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
;<br/>
<b>hence </b><a NAME="E9:42_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E1:42_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="altcat_4.html#E6:42_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="altcat_3.html#D7">ALTCAT_3:def 7</a></i>;<br/>


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

<b>assume </b><a NAME="E3:42"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="altcat_3.html#V5">epi</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D8">ALTCAT_3:def 8</a><br/>

<b>assume </b><a NAME="E4:42"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/>

<b>assume </b><a NAME="E5:42"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="altcat_2.html#T30">ALTCAT_2:30</a></i>;<br/>
<a NAME="E7:42"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E8:42"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <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="altcat_4.html#E4:42"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="txt" href="altcat_4.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_4.html#E4:42"><i><font color="Green">E29</font></i></a>, <a class="ref" href="altcat_2.html#T34">ALTCAT_2:34</a></i>;<br/>
<font color="Maroon">c<sub>13</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="altcat_4.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_4.html#E4:42"><i><font color="Green">E29</font></i></a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="altcat_4.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_4.html#E4:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_4.html#E5:42"><i><font color="Green">E30</font></i></a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
;<br/>
<b>hence </b><a NAME="E11:42"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E3:42"><i><font color="Green">E28</font></i></a>, <a class="txt" href="altcat_4.html#E8:42"><i><font color="Green">E31</font></i></a>, <a class="ref" href="altcat_3.html#D8">ALTCAT_3:def 8</a></i>;<br/>


</div>
