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<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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  implies  for <font color="Olive" title="b1">p</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="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="Olive" title="b1">p</font>)</span>)</span> holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</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="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="Olive" title="b1">p</font>)</span>)</span> holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font></span><br/>

<a NAME="E2:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/>
<a NAME="E3:7"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<a NAME="E4:7"/>
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="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<a NAME="E5:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">p</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="firebrick">::  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="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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> implies <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:7"/><span class="lab"><font color="Green" title="E9">A5</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="E7:7"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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">and </span><br/><a NAME="E8:7"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">g</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<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="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>);<br/>
<a NAME="E9:7"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
 not <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<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 class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="jordan1e.html#T5" target="_self" title="JORDAN1E:th.5">Th9</a></span>;<br/>
<a NAME="E10:7"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <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="#E7:7"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="jordan3.html#D3" title="JORDAN3:def.3">JORDAN3:def 3</a></span>;<br/>
<a NAME="E11:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A9</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>) <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> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<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 class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan3.html#D3" title="JORDAN3:def.3">JORDAN3:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1"/>
(  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<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="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<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="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><span class="kw">suppose </span></a><a NAME="E1:7_1_1"/>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<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="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:7_1_1"/><span class="kw">then </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>) <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="#E11:7"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/><a NAME="E3:7_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E4:7_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="topreal1.html#T22" title="TOPREAL1:th.22">TOPREAL1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="7_1_2"><span class="kw">suppose </span></a><a NAME="E1:7_1_2"/>
 not  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<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="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:7_1_2"/><span class="kw">then </span>
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<span class="p4">(<span class="default"><span class="p5">(<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="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<span class="p3">(<span class="default"><span class="p4">(<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="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="sppol_2.html#T20" title="SPPOL_2:th.20">SPPOL_2:20</a></span>;<br/><a NAME="E3:7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A10</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<span class="p3">(<span class="default"><span class="p4">(<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="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E9:7"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:7_1_2"/><span class="lab"><font color="Green" title="E15">A11</font></span>: 
1 <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="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/><a NAME="E5:7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A12</font></span>: 
2 <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="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:7_1_2"/><span class="lab"><font color="Green" title="E17">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E8:7_1_2"/><span class="lab"><font color="Green" title="E18">A14</font></span>: 
<font color="Maroon" title="c4">i</font> <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> <span class="p1">(<span class="default"><span class="p2"><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> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<span class="p3">(<span class="default"><span class="p4">(<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="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:7_1_2"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><span class="p2"><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> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<span class="p3">(<span class="default"><span class="p4">(<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="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E11:7"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="sppol_2.html#T13" title="SPPOL_2:th.13">SPPOL_2:13</a></span>;<br/><a NAME="E10:7_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><span class="p2"><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> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<span class="p3">(<span class="default"><span class="p4">(<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="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><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> <font color="Maroon" title="c4">i</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E7:7_1_2"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E8:7_1_2"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="ref" href="jordan3.html#T16" title="JORDAN3:th.16">JORDAN3:16</a></span>;<br/><a NAME="E11:7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<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> <font color="Maroon" title="c4">i</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E9:7_1_2"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="ref" href="jordan5b.html#T30" title="JORDAN5B:th.30">JORDAN5B:30</a></span>;<br/><a NAME="E12:7_1_2"/><span class="lab"><font color="Green" title="E21">A17</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="#E6:7"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="jordan3.html#T8" title="JORDAN3:th.8">JORDAN3:8</a></span>;<br/><a NAME="E13:7_1_2"/><span class="kw">then </span>
1 <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> <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> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_1_2"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E14:7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A18</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> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_1_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E15:7_1_2"/><span class="kw">then </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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_1_2"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E12:7_1_2"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="ref" href="jordan1e.html#T6" target="_self" title="JORDAN1E:th.6">Th10</a></span>;<br/><a NAME="E16:7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="jordan3.html#T9" title="JORDAN3:th.9">JORDAN3:9</a></span>;<br/><a NAME="E17:7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A19</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> 1</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">1 <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="#E4:7_1_2"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><a NAME="E18:7_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_1_2"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E12:7_1_2"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="txt" href="#E14:7_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="jordan1e.html#T6" target="_self" title="JORDAN1E:th.6">Th10</a></span>;<br/><a NAME="E19:7_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c1">f</font>,<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 class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <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="#E3:7"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E14:7_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E5:7_1_2"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="sprect_2.html#T8" title="SPRECT_2:th.8">SPRECT_2:8</a></span>;<br/><span class="kw">hence </span><a NAME="E20:7_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:7"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E3:7_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E17:7_1_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="ref" href="sprect_3.html#T6" title="SPRECT_3:th.6">SPRECT_3:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
