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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="altcat_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>, <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> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</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>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="altcat_3.html#V9" target="_self">_zero</a>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="altcat_3.html#V9" target="_self">_zero</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D12" target="_self">ALTCAT_3:def 12</a><br/>

<b>assume </b><a NAME="E3:51"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="altcat_3.html#V8" target="_self">_zero</a>
 ;<br/>

<a NAME="E4:51"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="altcat_3.html#V6" target="_self">initial</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="altcat_3.html#V7" target="_self">terminal</a> )
 
<b>by </b><i><a class="ref" href="altcat_3.html#D11" target="_self">Def11</a></i>;<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>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:51"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>10</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>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_3.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E8:51"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>11</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>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_3.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E10:51"/><i><font color="Green">E35</font></i>: 
( <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>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_3.html#D10" target="_self">Def10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E12:51"/><i><font color="Green">E36</font></i>: 
( <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>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_3.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E13:51"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</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>13</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:51"><i><font color="Green">E30</font></i></a>, <a class="txt" href="altcat_3.html#E3:51"><i><font color="Green">E31</font></i></a>, <a class="ref" href="altcat_3.html#D12" target="_self">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:51"/><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>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E10:51"><i><font color="Green">E35</font></i></a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<a NAME="E16:51"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E15:51"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:51"/><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>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E8:51"><i><font color="Green">E34</font></i></a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<a NAME="E19:51"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E18:51"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:51"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> st <br/>( <font color="Olive">b<sub>1</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>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_3.html#D10" target="_self">Def10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E22:51"/><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>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<a NAME="E23:51"/><i><font color="Green">E43</font></i>: 
(  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>13</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>16</sub></font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E22:51"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E24:51"/><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>7</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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</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_3.html#E6:51"><i><font color="Green">E33</font></i></a>, <a class="txt" href="altcat_3.html#E8:51"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_3.html#E10:51"><i><font color="Green">E35</font></i></a>, <a class="txt" href="altcat_3.html#E12:51"><i><font color="Green">E36</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</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_3.html#E1:51"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_3.html#E3:51"><i><font color="Green">E31</font></i></a>, <a class="txt" href="altcat_3.html#E13:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="altcat_3.html#D12" target="_self">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</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>10</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E24:51"><i><font color="Green">E44</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E6:51"><i><font color="Green">E33</font></i></a>, <a class="txt" href="altcat_3.html#E8:51"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_3.html#E12:51"><i><font color="Green">E36</font></i></a>, <a class="txt" href="altcat_3.html#E16:51"><i><font color="Green">E39</font></i></a>, <a class="txt" href="altcat_3.html#E19:51"><i><font color="Green">E41</font></i></a>, <a class="txt" href="altcat_3.html#E23:51"><i><font color="Green">E43</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</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_3.html#E8:51"><i><font color="Green">E34</font></i></a>, <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>
;<br/>


</div>
