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<b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">f2</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:15"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f1</font>,<font color="Maroon">f2</font> <a href="jordan12.html#R2" target="_self">are_in_general_position</a> 
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E2:15"/><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</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">f1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</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">f2</font> )
 ;<br/>

<a NAME="E3:15"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">f1</font> <a href="jordan12.html#R1" target="_self">is_in_general_position_wrt</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f2</font> <a href="jordan12.html#R1" target="_self">is_in_general_position_wrt</a> <font color="Maroon">f1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:15"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:15"/><i><font color="Green">E55</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E56</font></i>: 
not <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</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">f2</font>,<font color="Maroon">j</font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 ;<br/><b>set </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</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">f2</font>,<font color="Maroon">j</font></span>)</span>;<br/><b>set </b><font color="Maroon">A1</font> =  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font>;<br/><b>set </b><font color="Maroon">A2</font> =  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font>;<br/><b>set </b><font color="Maroon">B1</font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">B2</font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</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">f2</font>,<font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</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">f2</font>,<font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="realset1.html#T14">REALSET1:14</a></i>;<br/><a NAME="E4:15_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a1</font>, <font color="Maroon">a2</font> = <font color="Maroon">a2</font> as   <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> <b>by </b><i/>;<br/><a NAME="E6:15_1"/><i><font color="Green">E59</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E7:15_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E8:15_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E9:15_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:15_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">a2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T11">TOPREAL1:11</a></i>;<br/><a NAME="E11:15_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">a2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T11">TOPREAL1:11</a></i>;<br/><a NAME="E12:15_1"/>
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E13:15_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><a NAME="E14:15_1"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E15:15_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><a NAME="E16:15_1"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E17:15_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><a NAME="E18:15_1"/>
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E19:15_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><a NAME="E20:15_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E21:15_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E22:15_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E23:15_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">a2</font> or <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_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="E1:15_1_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">a2</font> or <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="jordan4.html#T53">JORDAN4:53</a></i>;<br/><a NAME="E3:15_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_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="E1:15_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font> )
 </span><b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="jordan4.html#T53">JORDAN4:53</a></i>;<br/><a NAME="E3:15_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_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="E1:15_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font> )
 </span><b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:15_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_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="E1:15_1_1_2_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">a2</font> or <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_2_1_1_1"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="jordan4.html#T53">JORDAN4:53</a></i>;<br/><a NAME="E3:15_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_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="E1:15_1_1_2_1_1_1_1_1"/>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font> )
 </span><b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2_1_1_2"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a2</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="jordan4.html#T53">JORDAN4:53</a></i>;<br/><a NAME="E3:15_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_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="E1:15_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font> or <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font> )
 </span><b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:15_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:15"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</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">f2</font>,<font color="Maroon">j</font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 ;<br/>


</div>
