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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:52"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V2">unfolded</a>
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V3">s.n.c.</a>
 <b>and </b><br/><a NAME="E3:52"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:52"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a><br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E5:52"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="sppol_2.html#E1:52_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1"><b>case </b></a><a NAME="E1:52_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_2"><b>case </b></a><a NAME="E1:52_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:52_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E4:52_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:52_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:52_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E3:52_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D4">FINSEQ_5:def 4</a></i>;<br/><a NAME="E4:52_1_2"/><i><font color="Green">E53</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:52_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E5:52_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:52_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:52"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:52_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_5.html#T72">FINSEQ_5:72</a></i>;<br/><a NAME="E8:52_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><a NAME="E9:52_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><a NAME="E10:52_1_2"/><i><font color="Green">E58</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E11:52_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E12:52_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:52_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:52_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E14:52_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:52"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E15:52_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E16:52_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="sppol_2.html#E7:52_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:52_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E18:52_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E19:52_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E17:52_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E11:52_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:52_1_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:52_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:52_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:52_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:52_1_2_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:52_1_2_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:52_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E11:52_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then </b><i><font color="Green">E68</font></i>:  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E6:52_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E6:52_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self">Th3</a></i>
;<br/> <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sppol_2.html#E2:52_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self">Th3</a></i>
;<br/><b>hence </b><a NAME="E10:52_1_2_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52"><i><font color="Green">E46</font></i></a>, <a class="txt" href="sppol_2.html#E5:52"><i><font color="Green">E49</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:52_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:52_1_2_1_1_2_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:52_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:52_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E5:52_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:52_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:52_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E12:52_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/><a NAME="E7:52_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:52_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>;<br/><a NAME="E9:52_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:52"><i><font color="Green">E47</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="topreal1.html#T11">TOPREAL1:11</a></i>;<br/><a NAME="E10:52_1_2_1_1_2_1_1_1"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><i><font color="Green">E73</font></i>:  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="sppol_2.html#E10:52_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E10:52_1_2_1_1_2_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
;<br/><a NAME="E12:52_1_2_1_1_2_1_1_1"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:52_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_5.html#T77">FINSEQ_5:77</a></i>;<br/><a NAME="E13:52_1_2_1_1_2_1_1_1"/><i><font color="Green">E75</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E14:52_1_2_1_1_2_1_1_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E7:52_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sppol_2.html#E10:52_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E15:52_1_2_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E16:52_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E17:52_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:52_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sppol_2.html#E12:52_1_2_1_1_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2_1_1_2_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="sppol_2.html#E14:52_1_2_1_1_2_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E18:52_1_2_1_1_2_1_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:52"><i><font color="Green">E47</font></i></a>, <a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal1.html#T14">TOPREAL1:14</a></i>;<br/><a NAME="E19:52_1_2_1_1_2_1_1_1"/><i><font color="Green">E79</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="sppol_2.html#E11:52_1_2_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E20:52_1_2_1_1_2_1_1_1"/><i><font color="Green">E80</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:52_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="sppol_2.html#E17:52_1_2_1_1_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:52_1_2_1_1_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E11:52_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then </b><i><font color="Green">E82</font></i>:  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self">Th3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52"><i><font color="Green">E46</font></i></a>, <a class="txt" href="sppol_2.html#E5:52"><i><font color="Green">E49</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E3:52_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E19:52_1_2_1_1_2_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E4:52_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:52_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E6:52_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>then </b><i><font color="Green">E86</font></i>:  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self">Th3</a></i>
;<br/><a NAME="E4:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E16:52_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:52"><i><font color="Green">E45</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><b>assume </b><a NAME="E6:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/>
not  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><a NAME="E9:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:52_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard2.html#T3">GOBOARD2:3</a></i>;<br/><a NAME="E10:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E12:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:52_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="sppol_2.html#E17:52_1_2_1_1_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sppol_2.html#E10:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E14:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E15:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E11:52_1_2_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E16:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="sppol_2.html#E11:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E17:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E11:52_1_2_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="sppol_2.html#E17:52_1_2_1_1_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="sppol_2.html#E18:52_1_2_1_1_2_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="sppol_2.html#E14:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E18:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:52"><i><font color="Green">E48</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52"><i><font color="Green">E46</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E4:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E20:52_1_2_1_1_2_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E5:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E6:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E7:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:52_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="sppol_2.html#E6:52_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E7:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E8:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><i><font color="Green">E87</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E9:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E10:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/> <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E7:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="sppol_2.html#T7" target="_self">Th7</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E12:52_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="sppol_2.html#T4" target="_self">Th4</a></i>
;<br/><b>hence </b><a NAME="E14:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52"><i><font color="Green">E46</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sppol_2.html#E12:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:52_1_2_1_1_2_1_1_1_2_1_1_2_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:52_1_2_1_1_2_1_1_1_2_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:52"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:52_1_2_1_1_2_1_1_1_2_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:52_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="sppol_2.html#E6:52_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="sppol_2.html#E5:52"><i><font color="Green">E49</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:52_1_2_1_1_2_1_1_1_2_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E7:52_1_2_1_1_2_1_1_1_2_1_2"/><i><font color="Green">E84</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E8:52_1_2_1_1_2_1_1_1_2_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><i><font color="Green">E86</font></i>:  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E6:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="sppol_2.html#E7:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="sppol_2.html#T7" target="_self">Th7</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E12:52_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="sppol_2.html#E7:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="sppol_2.html#T4" target="_self">Th4</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_2_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:52_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:52_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52"><i><font color="Green">E46</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E5:52_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E19:52_1_2_1_1_2_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E6:52_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E7:52_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E8:52_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="sppol_2.html#E7:52_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:52"><i><font color="Green">E45</font></i></a>, <a class="txt" href="sppol_2.html#E10:52_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><b>assume </b><a NAME="E4:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/>
not  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><a NAME="E7:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:52_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard2.html#T3">GOBOARD2:3</a></i>;<br/><a NAME="E8:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="sppol_2.html#E11:52_1_2_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E12:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sppol_2.html#E17:52_1_2_1_1_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E14:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E15:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E11:52_1_2_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="sppol_2.html#E17:52_1_2_1_1_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="sppol_2.html#E18:52_1_2_1_1_2_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="sppol_2.html#E12:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E16:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:52"><i><font color="Green">E48</font></i></a>, <a class="txt" href="sppol_2.html#E7:52_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E11:52_1_2_1_1_2_1_1_1_2_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_1_2_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:52_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:52_1_2_1_1_2_1_1_1_2_1_3"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:52_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="sppol_2.html#E6:52_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:52_1_2_1_1_2_1_1_1_2_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:52_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:52"><i><font color="Green">E49</font></i></a>, <a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E9:52_1_2_1_1_2_1_1_1_2_1_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E10:52_1_2_1_1_2_1_1_1_2_1_3"/><i><font color="Green">E83</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:52"><i><font color="Green">E49</font></i></a>, <a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E11:52_1_2_1_1_2_1_1_1_2_1_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><i><font color="Green">E85</font></i>:  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sppol_2.html#E10:52_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green">E83</font></i></a>, <a class="ref" href="sppol_2.html#T7" target="_self">Th7</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E12:52_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="sppol_2.html#E10:52_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="sppol_2.html#E11:52_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green">E84</font></i></a>, <a class="ref" href="sppol_2.html#T4" target="_self">Th4</a></i>
;<br/><a NAME="E13:52_1_2_1_1_2_1_1_1_2_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:52"><i><font color="Green">E49</font></i></a>, <a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E14:52_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52"><i><font color="Green">E46</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E15:52_1_2_1_1_2_1_1_1_2_1_3"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:52_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sppol_2.html#E9:52_1_2_1_1_2_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="sppol_2.html#E17:52_1_2_1_1_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E16:52_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sppol_2.html#E12:52_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E17:52_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="sppol_2.html#E14:52_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E18:52_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E19:52_1_2_1_1_2_1_1_1_2_1_3"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E22:52_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2_1_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E4:52_1_2_1_1_2_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="sppol_2.html#E4:52_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:52_1_2_1_1_2_1_1_2"/><i><font color="Green">E69</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52_1_2_1_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E7:52_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E8:52_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="sppol_2.html#E19:52_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:52_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:52_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E11:52_1_2_1_1_2_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><i><font color="Green">E72</font></i>:  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E4:52_1_2_1_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sppol_2.html#E6:52_1_2_1_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="sppol_2.html#T7" target="_self">Th7</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E6:52_1_2_1_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sppol_2.html#E10:52_1_2_1_1_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sppol_2.html#E11:52_1_2_1_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="sppol_2.html#T5" target="_self">Th5</a></i>
;<br/><a NAME="E13:52_1_2_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:52_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="sppol_2.html#E6:52_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E15:52_1_2_1_1_2_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:52_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="sppol_2.html#E1:52_1_2_1_1_2_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E17:52_1_2_1_1_2_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E18:52_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E15:52_1_2_1_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:52_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E20:52_1_2_1_1_2_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E21:52_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:52"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E22:52_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/> <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E3:52_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sppol_2.html#E8:52_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sppol_2.html#E13:52_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sppol_2.html#E17:52_1_2_1_1_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="sppol_2.html#E19:52_1_2_1_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="sppol_2.html#T7" target="_self">Th7</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="sppol_2.html#E12:52_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="sppol_2.html#E19:52_1_2_1_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="sppol_2.html#E20:52_1_2_1_1_2_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="sppol_2.html#T4" target="_self">Th4</a></i>
;<br/><b>hence </b><a NAME="E24:52_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:52"><i><font color="Green">E46</font></i></a>, <a class="txt" href="sppol_2.html#E12:52_1_2_1_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="sppol_2.html#E22:52_1_2_1_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:52_1_2_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E21:52_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5">Ins</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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