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<b>set </b><font color="Maroon">Cr</font> = 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>;<br/>
<a NAME="E1:22_1_1"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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></span>)</span>,<span class="p2">(<span class="default">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></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/>
<b>consider </b><font color="Maroon">Ar</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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></span>)</span>,<span class="p2">(<span class="default">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></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E3:22_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">Ar</font> <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>and </b><br/><a NAME="E4:22_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="yellow20.html#K1" target="_self">Intersect</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font></span><a href="altcat_1.html#K2">|}</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">Ar</font></span><a href="altcat_1.html#K2">|}</a></span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Ar1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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></span>)</span>,<span class="p2">(<span class="default">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></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">Ar2</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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></span>)</span>,<span class="p2">(<span class="default">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></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E6:22_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">Ar1</font> <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; <font color="Maroon">Ar2</font> <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>and </b><br/><a NAME="E7:22_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="yellow20.html#K1" target="_self">Intersect</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">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">A</font></span><a href="altcat_1.html#K3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">Ar1</font>,<font color="Maroon">Ar2</font></span><a href="altcat_1.html#K3">|}</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E8:22_1_1"/>
( <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">A</font>,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#K3">:]</a></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="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>,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="zfmisc_1.html#K3">[:</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K3">:]</a></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="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>,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#D3">ZFMISC_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E52</font></i>: <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">A</font>,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#K3">:]</a></span> <a href="xboole_0.html#K3">/\</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">B</font>,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#K3">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default">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></span>)</span>,<span class="p3">(<span class="default">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></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p2">(<span class="default">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></span>)</span></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/>.= 
<span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default">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></span>)</span>,<span class="p2">(<span class="default">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></span>)</span>,<span class="p2">(<span class="default">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></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#D3">ZFMISC_1:def 3</a></i>
;<br/>
<a NAME="E10:22_1_1"/><i><font color="Green">E53</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</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">A</font>,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#K3">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</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">B</font>,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#K3">:]</a></span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_1_2"/>
( <font color="Maroon">a1</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">a2</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">a3</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E4:22_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T72">MCART_1:72</a></i>;<br/><b>assume </b><a NAME="E5:22_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font>
 ;<br/><b>thus </b><a NAME="E6:22_1_1_2"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="partfun1.html#R1">tolerates</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Cm</font> =  <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> as    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">Ar</font>,<font color="Maroon">Ar</font></span><a href="altcat_1.html#K3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">Ar</font></span><a href="altcat_1.html#K2">|}</a></span> <b>by </b><i>, <a class="ref" href="yellow20.html#T1" target="_self">Th1</a>, , , , </i>;<br/>
<b>take </b>
 <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default">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></span>)</span>,<font color="Maroon">Ar</font>,<font color="Maroon">Cm</font> #)
;<br/>

<b>thus </b><a NAME="E13:22_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default">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></span>)</span>,<font color="Maroon">Ar</font>,<font color="Maroon">Cm</font> #) <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 <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default">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></span>)</span>,<font color="Maroon">Ar</font>,<font color="Maroon">Cm</font> #) <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 <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default">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></span>)</span>,<font color="Maroon">Ar</font>,<font color="Maroon">Cm</font> #) <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/>


</div>
