<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  holds  <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>



<a NAME="E1:3"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="jordan3.html#K3" title="JORDAN3:func.3">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">L_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1j.html#T44" title="JORDAN1J:th.44">JORDAN1J:44</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:3_1"/>
( ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> ) or (  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><span class="kw">suppose </span></a><a NAME="E1:3_1_1"/>
( ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> ) or (  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:3_1_1"/><span class="kw">then </span>
 <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K3" title="JORDAN3:func.3">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">L_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan3.html#D8" title="JORDAN3:def.8">JORDAN3:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:3_1_1"/>
 <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1j.html#T44" title="JORDAN1J:th.44">JORDAN1J:44</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2"><span class="kw">suppose </span></a><a NAME="E1:3_1_2"/>
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:3_1_2"/><span class="kw">then </span>
 <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3" title="JORDAN3:func.3">R_Cut</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">L_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan3.html#D8" title="JORDAN3:def.8">JORDAN3:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:3_1_2"/>
 <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E3">A1</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
