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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#V2">transitive</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">D</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" target="_self">SubCatStr</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">D</font>, <font color="Maroon">p2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">D</font>, <font color="Maroon">p3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><a NAME="E1:70"/><i><font color="Green">E16</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</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">p2</font>,<font color="Maroon">p3</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">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E2:70"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> )
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o3</font>;<br/><b>let </b><font color="Maroon">ff</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">p1</font>,<font color="Maroon">p2</font>;<br/><b>let </b><font color="Maroon">gg</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">p2</font>,<font color="Maroon">p3</font>;<br/>







<b>assume </b><a NAME="E3:70"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font> )
 ;<br/>

<a NAME="E4:70"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</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="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E5:70"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">D</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</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">&lt;^</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="altcat_1.html#K1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="altcat_1.html#K1">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:70"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">gg</font>,<font color="Maroon">ff</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">D</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:70"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="domain_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">D</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">D</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T73">MCART_1:73</a></i>;<br/>
<a NAME="E8:70"/><i><font color="Green">E29</font></i>: 
( the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">D</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="domain_1.html#K4">]</a></span> &amp; the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="domain_1.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E9:70"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">D</font> <a href="altcat_2.html#R2" target="_self">cc=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E3:70"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E10:70"/><b>then </b><i><font color="Green">E30</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">D</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E11:70"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</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">o1</font>,<font color="Maroon">o2</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">p2</font>,<font color="Maroon">p3</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">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:70"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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">o2</font>,<font color="Maroon">o3</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="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</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="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">D</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">gg</font>,<font color="Maroon">ff</font>
<b>by </b><i>, , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">gg</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">ff</font>
<b>by </b><i>, <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
;<br/><br/>


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