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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="finseq_6.html#V1" title="FINSEQ_6:attr.1">circular</a>   <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 4 implies for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> implies <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">j</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:37"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">and </span><br/><a NAME="E3:37"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font>
 <span class="kw">and </span><br/><a NAME="E4:37"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">and </span><br/><a NAME="E5:37"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:37"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E7:37"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
( <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E8:37"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></span>;<br/>
<a NAME="E10:37"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:37"/><span class="kw">then </span>
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E12:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_1"/>
( ( <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or ( <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) or ( <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> ) or ( <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:37_1_1"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font>
 <span class="kw">and </span><br/><a NAME="E2:37_1_1"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E3:37_1_1"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:37_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/><a NAME="E5:37_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A14</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E6:37_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E7:37_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a></span>;<br/><a NAME="E8:37_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E9:37_1_1"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></span>;<br/><a NAME="E10:37_1_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:37_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E12:37_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E9:37_1_1"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E13:37_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:37_1_2"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font>
 <span class="kw">and </span><br/><a NAME="E2:37_1_2"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E3:37_1_2"/><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E4:37_1_2"/>
<font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E1:37_1_2"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="txt" href="#E2:37_1_2"><span class="lab"><font color="Green" title="E41">A18</font></span></a></span>;<br/><a NAME="E5:37_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E3:37_1_2"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a></span>;<br/><a NAME="E6:37_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A20</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E7:37_1_2"/>
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E2:37_1_2"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E3:37_1_2"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E8:37_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E6:37_1_2"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_3"><span class="kw">suppose </span></a><a NAME="E1:37_1_3"/><span class="lab"><font color="Green" title="E40">A21</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:37_1_3"/><span class="lab"><font color="Green" title="E41">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E3:37_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A23</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_3"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:37_1_3"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E2:37_1_3"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a></span>;<br/><a NAME="E5:37_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A24</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E6:37_1_3"/>
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E2:37_1_3"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="txt" href="#E3:37_1_3"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E7:37_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E5:37_1_3"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:37_1_4"/><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font>
 <span class="kw">and </span><br/><a NAME="E2:37_1_4"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E3:37_1_4"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E2:37_1_4"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:37_1_4"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E1:37_1_4"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a></span>;<br/><a NAME="E5:37_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:37_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E12:37"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_5"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:37_1_5"/><span class="lab"><font color="Green" title="E40">A27</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font>
 <span class="kw">and </span><br/><a NAME="E2:37_1_5"/><span class="lab"><font color="Green" title="E41">A28</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E3:37_1_5"/>
<font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E2:37_1_5"><span class="lab"><font color="Green" title="E41">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:37_1_5"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_5"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a></span>;<br/><a NAME="E5:37_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:37_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E12:37"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_6"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:37_1_6"/><span class="lab"><font color="Green" title="E40">A29</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font>
 <span class="kw">and </span><br/><a NAME="E2:37_1_6"/><span class="lab"><font color="Green" title="E41">A30</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E3:37_1_6"/><span class="lab"><font color="Green" title="E42">A31</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E4:37_1_6"/><span class="lab"><font color="Green" title="E43">A32</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_6"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:37_1_6"/><span class="lab"><font color="Green" title="E44">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:37_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A34</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_6"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:37_1_6"/>
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E2:37_1_6"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E3:37_1_6"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="txt" href="#E5:37_1_6"><span class="lab"><font color="Green" title="E44">A33</font></span></a></span>;<br/><a NAME="E8:37_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_6"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E9:37_1_6"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,1 <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_6"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E5:37_1_6"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="txt" href="#E4:37_1_6"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a></span>;<br/><a NAME="E10:37_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A35</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E11:37_1_6"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_6"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E6:37_1_6"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:37_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E5:37_1_6"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E13:37_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E12:37"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E2:37_1_6"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E10:37_1_6"><span class="lab"><font color="Green" title="E46">A35</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
