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

<b>assume </b><a NAME="E1:25"/><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/>

<b>let </b><font color="Maroon">a1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">a2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>, <font color="Maroon">b2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>;<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>;<br/>





<b>assume </b><a NAME="E2:25"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 ;<br/>

<a NAME="E3:25"/><i><font color="Green">E33</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> )
 
<b>by </b><i><a class="ref" href="yellow20.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E4:25"/><b>then </b><i><font color="Green">E34</font></i>: 
<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> <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 <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/>
<a NAME="E5:25"/><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <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 <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <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 <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <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/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E2:25_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:25_1"/>
<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> <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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E2:25_2"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><a NAME="E3:25_2"/><b>then </b>
( <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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E4:25_2"/>
( <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &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="funct_1.html#K1">.</a> <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#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E8:25"/>
<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>
 <b>by </b><i>, , , , , <a class="txt" href="yellow20.html#E1:25"><i><font color="Green">E25</font></i></a>, </i>;<br/>


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