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<span class="kw">let </span><font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c2">J</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><br/> for <font color="Olive" title="b3">G1</font>, <font color="Olive" title="b4">G2</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  ex <font color="Olive" title="b5">H1</font>, <font color="Olive" title="b6">H2</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b5">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b3">G1</font> &amp; <font color="Olive" title="b6">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b4">G2</font> &amp;  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Olive" title="b3">G1</font>,<font color="Olive" title="b4">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Olive" title="b5">H1</font>,<font color="Olive" title="b6">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  ex <font color="Olive" title="b3">H1</font>, <font color="Olive" title="b4">H2</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b3">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c3">F1</font>,<font color="Olive" title="b1">G1</font> &amp; <font color="Olive" title="b4">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c4">F2</font>,<font color="Olive" title="b2">G2</font> &amp;  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Olive" title="b3">H1</font>,<font color="Olive" title="b4">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">G1</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c6">G2</font> &amp;  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> )</span><br/>





<a NAME="E1:21"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E2:21"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">H1</font> =  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c8">H2</font> =  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c6">G2</font> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow20.html#T14" target="_self" title="YELLOW20:th.14">Th14</a></span>;<br/>
<a NAME="E4:21"/>
( <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D3" title="ZFMISC_1:def.3">ZFMISC_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E17">A3</font></span>: <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></span>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D3" title="ZFMISC_1:def.3">ZFMISC_1:def 3</a></span>
;<br/>
<a NAME="E6:21"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E19">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> implies <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c7">H1</font>,<font color="Maroon" title="c8">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:21_2"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c7">H1</font>,<font color="Maroon" title="c8">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><br/><a NAME="E2:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:21_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font>, <font color="Maroon" title="c12">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:21_2"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 <span class="kw">and </span><br/><a NAME="E6:21_2"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 <span class="kw">and </span><br/><a NAME="E7:21_2"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 <span class="kw">and </span><br/><a NAME="E8:21_2"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T72" title="MCART_1:th.72">MCART_1:72</a></span>;<br/><a NAME="E9:21_2"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="mcart_1.html#T73" title="MCART_1:th.73">MCART_1:73</a></span>;<br/><a NAME="E10:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E11:21_2"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="mcart_1.html#T73" title="MCART_1:th.73">MCART_1:73</a></span>;<br/><a NAME="E12:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E13:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_2"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E14:21_2"/><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="mcart_1.html#T73" title="MCART_1:th.73">MCART_1:73</a></span>;<br/><a NAME="E15:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_2"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E16:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_2"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E17:21_2"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_2"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E6:21_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E7:21_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></span>;<br/><a NAME="E18:21_2"/><span class="lab"><font color="Green" title="E35">A21</font></span>: 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_2"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="txt" href="#E9:21_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E11:21_2"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></span>;<br/><a NAME="E19:21_2"/>
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c7">H1</font>,<font color="Maroon" title="c8">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">H2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">H1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_2"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E10:21_2"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E12:21_2"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></span>;<br/><span class="kw">hence </span><span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c7">H1</font>,<font color="Maroon" title="c8">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">H2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">H1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></span>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">H1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E13:21_2"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="yellow20.html#D2" target="_self" title="YELLOW20:def.2">Def2</a></span>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E16:21_2"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="ref" href="yellow20.html#D2" target="_self" title="YELLOW20:def.2">Def2</a></span>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E17:21_2"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E18:21_2"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">take </span>
<font color="Maroon" title="c7">H1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">H2</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Maroon" title="c7">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">G1</font> &amp; <font color="Olive" title="b1">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c6">G2</font> &amp;  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c7">H1</font>,<font color="Olive" title="b1">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">H2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c8">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c6">G2</font> &amp;  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c7">H1</font>,<font color="Maroon" title="c8">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E8:21"/>
( <font color="Maroon" title="c7">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c8">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c6">G2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c7">H1</font>,<font color="Maroon" title="c8">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span></span><br/>

<a NAME="E9:21"/><span class="lab"><font color="Green" title="E20">A22</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c7">H1</font>,<font color="Maroon" title="c8">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">J</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E10:21"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:21"/>
 <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c7">H1</font>,<font color="Maroon" title="c8">H2</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="yellow20.html#D2" target="_self" title="YELLOW20:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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