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<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"> ( <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="jordan23.html#V2" title="JORDAN23:attr.2">weakly-one-to-one</a>  implies 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>  st <font color="Olive" title="b1">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="Olive" title="b2">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> holds <br/> <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:53"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
( <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="jordan23.html#V2" title="JORDAN23:attr.2">weakly-one-to-one</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">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>  st <font color="Olive" title="b1">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="Olive" title="b2">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> holds <br/> <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</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"> ( <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> implies  <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:53"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:53"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<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>
 ; <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> </span><br/>

<a NAME="E4:53"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
 <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="jordan3.html#T41" title="JORDAN3:th.41">JORDAN3:41</a></span>;<br/>
<a NAME="E5:53"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="jordan3.html#T41" title="JORDAN3:th.41">JORDAN3:41</a></span>;<br/>
<a NAME="E6:53"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 <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#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_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="E1:53_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font> or <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><span class="kw">suppose </span></a><a NAME="E1:53_1_1"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 ; <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> </span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_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="E1:53_1_1_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> ) 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="53_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
( <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> )
 ; <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> </span><br/><div class="add"><a NAME="E2:53_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<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> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan3.html#T64" title="JORDAN3:th.64">JORDAN3:64</a></span>;<br/><a NAME="E3:53_1_1_1_1_1"/>
 <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="jordan23.html#T29" target="_self" title="JORDAN23:th.29">Th29</a></span>;<br/><a NAME="E4:53_1_1_1_1_1"/><span class="kw">then </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="c2">p</font></span>)</span>,<font color="Maroon" title="c3">q</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="jordan23.html#T30" target="_self" title="JORDAN23:th.30">Th30</a></span>;<br/><span class="kw">hence </span><a NAME="E5:53_1_1_1_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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="jordan3.html#D8" title="JORDAN3:def.8">JORDAN3:def 8</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="53_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
(  <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> </span><br/><div class="add"><a NAME="E2:53_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
 <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="jordan23.html#T29" target="_self" title="JORDAN23:th.29">Th29</a></span>;<br/><a NAME="E3:53_1_1_1_1_2"/>
<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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E1:53_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E1:53_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="jordan3.html#T66" title="JORDAN3:th.66">JORDAN3:66</a></span>;<br/><a NAME="E4:53_1_1_1_1_2"/><span class="kw">then </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="c2">p</font></span>)</span>,<font color="Maroon" title="c3">q</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="jordan23.html#T30" target="_self" title="JORDAN23:th.30">Th30</a></span>;<br/><span class="kw">hence </span><a NAME="E5:53_1_1_1_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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="jordan3.html#D8" title="JORDAN3:def.8">JORDAN3:def 8</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="53_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_1_1_3"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
( 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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> </span><br/><div class="add"><div><span class="lab"><font color="Green" title="E35">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_3_1_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="E1:53_1_1_1_1_3_1_1"/>
(  <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> <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="c2">p</font>,<font color="Maroon" title="c1">f</font> or  <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> <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="c2">p</font>,<font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E1:53_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_1_1_3_1_1_1"/>
 <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> <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="c2">p</font>,<font color="Maroon" title="c1">f</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="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> <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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:53_1_1_1_1_3_1_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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="jordan3.html#T64" title="JORDAN3:th.64">JORDAN3:64</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="53_1_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
 <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> <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="c2">p</font>,<font color="Maroon" title="c1">f</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="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> <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></span><br/><div class="add"><a NAME="E2:53_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:53_1_1_1_1_3_1_1_2"/>
<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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:53_1_1_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A16</font></span>: 
<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> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">Ls</font> =  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <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></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<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>)</span>;<br/><a NAME="E5:53_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <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></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<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>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E1:53_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="jordan5b.html#T32" title="JORDAN5B:th.32">JORDAN5B:32</a></span>;<br/><a NAME="E6:53_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E40">A18</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <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></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<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>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="jordan5b.html#T32" title="JORDAN5B:th.32">JORDAN5B:32</a></span>;<br/><a NAME="E7:53_1_1_1_1_3_1_1_2"/>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="jordan23.html#D2" target="_self" title="JORDAN23:def.2">Def2</a></span>;<br/><a NAME="E8:53_1_1_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E9:53_1_1_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A19</font></span>: 
<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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E10:53_1_1_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A20</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <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></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<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>)</span> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <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 class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></span>;<br/><a NAME="E11:53_1_1_1_1_3_1_1_2"/>
not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>, <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <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></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<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>)</span>,<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E1:53_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E9:53_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="ref" href="jordan5c.html#T17" title="JORDAN5C:th.17">JORDAN5C:17</a></span>;<br/><a NAME="E12:53_1_1_1_1_3_1_1_2"/><span class="kw">then </span>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font>, <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <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></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<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>)</span>,<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E10:53_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E6:53_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="txt" href="#E5:53_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="jordan5c.html#T14" title="JORDAN5C:th.14">JORDAN5C:14</a></span>;<br/><span class="kw">hence </span><a NAME="E13:53_1_1_1_1_3_1_1_2"/>
<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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E1:53_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E1:53_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E9:53_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="ref" href="jordan3.html#T66" title="JORDAN3:th.66">JORDAN3:66</a>, <a class="ref" href="jordan5c.html#T17" title="JORDAN5C:th.17">JORDAN5C:17</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><span class="kw">end;</span></div><a NAME="E3:53_1_1_1_1_3"/><span class="lab"><font color="Green" title="E36">A21</font></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="txt" href="#E1:53_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="jordan3.html#D8" title="JORDAN3:def.8">JORDAN3:def 8</a></span>;<br/><a NAME="E4:53_1_1_1_1_3"/>
 <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="jordan23.html#T29" target="_self" title="JORDAN23:th.29">Th29</a></span>;<br/><a NAME="E5:53_1_1_1_1_3"/><span class="kw">then </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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1_1_1_1_3"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="jordan23.html#T30" target="_self" title="JORDAN23:th.30">Th30</a></span>;<br/><span class="kw">hence </span><a NAME="E6:53_1_1_1_1_3"/>
 <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53_1_1_1_1_3"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="sppol_2.html#T42" title="SPPOL_2:th.42">SPPOL_2:42</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:53_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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 ; <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="53_1_2"><span class="kw">suppose </span></a><a NAME="E1:53_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 ; <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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> </span><br/><div class="add"><a NAME="E2:53_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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="jordan23.html#T15" target="_self" title="JORDAN23:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:53_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> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T39" title="SPPOL_2:th.39">SPPOL_2:39</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>
