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

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_2.html#M1" target="_self">SubCatStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:71_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="altcat_1.html#V2">transitive</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_2.html#M1" target="_self">SubCatStr</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:71_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="altcat_1.html#V11">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>3</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D8" target="_self">ALTCAT_2:def 8</a><br/>

<a NAME="E1:71_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<a NAME="E2:71_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11" target="_self">Def11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</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="txt" href="altcat_2.html#E1:71_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>13</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>let </b><font color="Maroon">c<sub>14</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>7</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E4:71_1_1_1"/><i><font color="Green">E35</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>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> 
 <b>and </b><br/><a NAME="E5:71_1_1_1"/><i><font color="Green">E36</font></i>: 
<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> 
 <b>and </b><br/><a NAME="E6:71_1_1_1"/><i><font color="Green">E37</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>7</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/>

<a NAME="E7:71_1_1_1"/><i><font color="Green">E38</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>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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="ref" href="altcat_2.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E8:71_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>8</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> 
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:71_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E9:71_1_1_1"/><i><font color="Green">E40</font></i>: 
<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="tarski.html#R1">c=</a> <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>10</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="ref" href="altcat_2.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E10:71_1_1_1"/><b>then </b><i><font color="Green">E41</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>10</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_2.html#E5:71_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E11:71_1_1_1"/><i><font color="Green">E42</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>7</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>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="ref" href="altcat_2.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E12:71_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</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_2.html#E6:71_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E13:71_1_1_1"/><i><font color="Green">E44</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>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> 
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:71_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="altcat_2.html#E5:71_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E14:71_1_1_1"/><i><font color="Green">E45</font></i>: 
<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>7</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_2.html#E5:71_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="altcat_2.html#E6:71_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E15:71_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</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>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:71_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="altcat_2.html#E7:71_1_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E17:71_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <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>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E5:71_1_1_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="altcat_2.html#E9:71_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E19:71_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <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>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E6:71_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="altcat_2.html#E11:71_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E21:71_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:71_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="altcat_2.html#E5:71_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="altcat_2.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E22:71_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E5:71_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="altcat_2.html#E6:71_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="altcat_2.html#T33" target="_self">Th33</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="altcat_2.html#E4:71_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="altcat_2.html#E14:71_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="altcat_2.html#T33" target="_self">Th33</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_2.html#E8:71_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="altcat_2.html#E10:71_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="altcat_2.html#E12:71_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="altcat_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_2.html#E6:71_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="altcat_2.html#E13:71_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="altcat_2.html#E21:71_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="altcat_2.html#T33" target="_self">Th33</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:71_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="altcat_1.html#V11">associative</a>
 ;<br/>


</div>
