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<b>let </b><font color="Maroon">C</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V10" target="_self">pseudo-functional</a>  <a href="altcat_1.html#L2" target="_self">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">a2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">a3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green">E15</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="altcat_1.html#K1" target="_self">^&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" target="_self">&lt;^</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="altcat_1.html#K1" target="_self">^&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" target="_self">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:40"/><i><font color="Green">E27</font></i>: 
the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:40"/><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font></span>)</span>,<span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p3"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">a2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a2</font>,<font color="Maroon">a3</font>;<br/><b>let </b><font color="Maroon">f'</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g'</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>





<b>assume </b><a NAME="E4:40"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> )
 ;<br/>

<a NAME="E5:40"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:40"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:40"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:40"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">f''</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g''</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:40"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></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"><font color="Maroon">g''</font>,<font color="Maroon">f''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:40"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g''</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f''</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E3:40"><i><font color="Green">E28</font></i></a>, <a class="txt" href="altcat_1.html#E4:40"><i><font color="Green">E29</font></i></a>, , </i>;<br/>
<a NAME="E10:40"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g''</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f''</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>thus </b><font color="Maroon">g</font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:40"><i><font color="Green">E27</font></i></a>, </i>
<br/>.= 
<font color="Maroon">g'</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f'</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E3:40"><i><font color="Green">E28</font></i></a>, , , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>


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