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<b>let </b><font color="Maroon">A</font> be  <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_1.html#L2">AltCatStr</a> , <font color="Maroon">B</font> be  <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/>

<b>set </b><font color="Maroon">AB</font> =  <a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>
<b>assume </b><a NAME="E1:26"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="yellow20.html#R1" target="_self">have_the_same_composition</a> 
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E32</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow20.html#K1" target="_self">Intersect</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> &amp; the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow20.html#K1" target="_self">Intersect</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>, <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a><br/>

<b>assume </b><a NAME="E3:26"/><i><font color="Green">E33</font></i>: 
( <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>  )
 ;<br/>

<a NAME="E4:26"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:26"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E6:26"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">o3</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" target="_self">Intersect</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:26"/><b>then </b>
( <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">o3</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Maroon">o3</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font>, <font color="Maroon">B</font> = <font color="Maroon">B</font> as  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>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">o1</font>, <font color="Maroon">a2</font> = <font color="Maroon">o2</font>, <font color="Maroon">a3</font> = <font color="Maroon">o3</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">o1</font>, <font color="Maroon">b2</font> = <font color="Maroon">o2</font>, <font color="Maroon">b3</font> = <font color="Maroon">o3</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font> <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:26"/><i><font color="Green">E51</font></i>: 
( <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#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</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">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">b3</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">o1</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:26"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</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">a2</font>,<font color="Maroon">a3</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">b1</font>,<font color="Maroon">b2</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">b2</font>,<font color="Maroon">b3</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/>;<br/>
<a NAME="E13:26"/><b>then </b><i><font color="Green">E52</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a3</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">b1</font>,<font color="Maroon">b3</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/>
<b>consider </b><font color="Maroon">f</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>, <font color="Maroon">g</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o3</font>;<br/>
<a NAME="E15:26"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">f</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">a1</font>,<font color="Maroon">a2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">f</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">b1</font>,<font color="Maroon">b2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">g</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">a2</font>,<font color="Maroon">a3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">g</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">b2</font>,<font color="Maroon">b3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i>, <a class="txt" href="yellow20.html#E1:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <b>by </b><i>, <a class="txt" href="yellow20.html#E1:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <b>by </b><i>, <a class="txt" href="yellow20.html#E1:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a2</font>,<font color="Maroon">a3</font> <b>by </b><i>, <a class="txt" href="yellow20.html#E1:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g2</font> = <font color="Maroon">g</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b2</font>,<font color="Maroon">b3</font> <b>by </b><i>, <a class="txt" href="yellow20.html#E1:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E20:26"/>
<font color="Maroon">g1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f2</font>
 
<b>by </b><i>, <a class="ref" href="yellow20.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>hence </b><a NAME="E21:26"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</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="txt" href="yellow20.html#E1:26"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
