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<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="struct_0.html#NM1">Element</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="funct_1.html#NM1">Function</a>;<br/>



<b>given </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>9</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>8</sub></font><b> such that </b><a NAME="E1:87_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &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>8</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="E2:87_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</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>  holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>given </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <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> such that </b><a NAME="E3:87_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <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>and </b><br/><a NAME="E4:87_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</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>  holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>13</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>11</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E1:87_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yellow18.html#E3:87_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>9</sub></font>;<br/>

<b>thus </b><a NAME="E6:87_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &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>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="yellow18.html#E1:87_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yellow18.html#E3:87_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E7:87_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>15</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="E8:87_1"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>8</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="yellow18.html#E1:87_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yellow18.html#E7:87_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>

<a NAME="E9:87_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:87_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="yellow18.html#E7:87_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E10:87_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yellow18.html#E9:87_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yellow18.html#E1:87_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yellow18.html#E3:87_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="yellow18.html#E4:87_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="yellow18.html#E8:87_1"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yellow18.html#E1:87_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yellow18.html#E3:87_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="yellow18.html#E7:87_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
;<br/><br/>


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