<?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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">and </span><br/><a NAME="E2:22"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E3:22"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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#V2" title="TOPREAL1:attr.2">unfolded</a> </span><br/>

<a NAME="E4:22"/>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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> implies <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">and </span><br/><a NAME="E2:22_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:22_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E21">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="E4:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A7</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="#E2:22_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E5:22_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="#E3:22_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E6:22_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
;<br/>
<a NAME="E7:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="#E2:22_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E8:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A10</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c2">i</font></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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E9:22_1"/>
( <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <font color="Maroon" title="c2">i</font></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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>) &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<a NAME="E10:22_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E11:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A11</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E12:22_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
<font color="Maroon" title="c2">i</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="#E7:22_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_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="E1:22_1_1"/>
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_1_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:22_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E11:22_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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="22_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_1_1_2"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_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="E1:22_1_1_2_1_1"/>
( <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:22_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:22_1_1_2_1_1_1"/><span class="kw">then </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> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2 &amp; <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> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="nat_d.html#T39" title="NAT_D:th.39">NAT_D:39</a></span>;<br/><span class="kw">hence </span><a NAME="E3:22_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E11:22_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E1:22_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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="22_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:22_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:22_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E1:22_1_1_2"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:22_1_1_2_1_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> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:22_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A16</font></span>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E5:22_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="#E2:22_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E1:22_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:22_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A17</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:22_1_1_2_1_1_2_1"/>
( <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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>,<font color="Maroon" title="c2">i</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/><a NAME="E2:22_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:22_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><span class="kw">assume </span><a NAME="E4:22_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:22_1_1_2_1_1_2_1"/><span class="kw">then </span>
 not <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:22_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:22_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
 not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E9:22_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E37">A21</font></span>: 
 <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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E5:22_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E6:22_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><a NAME="E10:22_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E38">A22</font></span>: 
<font color="Maroon" title="c3">x</font> <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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_1_1_2_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="E1:22_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) or <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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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> <font color="Maroon" title="c2">i</font></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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E7:22_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="txt" href="#E10:22_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="jgraph_1.html#T16" target="_self" title="JGRAPH_1:th.16">Th20</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_1_1_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:22_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A23</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 ; <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="E2:22_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A24</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E3:22_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:22_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></span>;<br/><a NAME="E5:22_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A25</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E6:22_1_1_2_1_1_2_1_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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="#E12:22_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E4:22_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E2:22_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><a NAME="E7:22_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <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="#E2:22_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E8:22_1_1_2_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E1:22_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="txt" href="#E5:22_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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="22_1_1_2_1_1_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:22_1_1_2_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A26</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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> <font color="Maroon" title="c2">i</font></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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)
 ; <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="E2:22_1_1_2_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:22_1_1_2_1_1_2_1_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="#E1:22"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></span>;<br/><a NAME="E4:22_1_1_2_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A27</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E5:22_1_1_2_1_1_2_1_1_1_2"/>
<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"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="#E9:22_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E6:22_1_1_2_1_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E1:22_1_1_2_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E4:22_1_1_2_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:22_1_1_2_1_1_2_1"/>
contradiction
 ; <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><span class="kw">hence </span><a NAME="E8:22_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:22_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:22"/>
<font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 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>
