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<div class="add">

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b2">n</font> &amp; not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> holds <br/> <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">p</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a></span><br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">n</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> holds <br/> <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">p</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a></span><br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> implies  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>
 <b>and </b><br/><a NAME="E4:50"/><i><font color="Green" title="E47">A4</font></i>: 
not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a></span><br/>

<b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font> )</span><br/>

<b>assume </b><a NAME="E5:50"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/>

<b>set </b><font color="Maroon" title="c6">fp</font> =  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span> or ( <font color="Maroon" title="c4">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>  &amp; <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/><i><font color="Green" title="E49">A6</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_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="E1:50_1_1_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="#E1:50_1_1"><i><font color="Green" title="E49">A6</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_1"><b>case </b></a><a NAME="E1:50_1_1_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:50_1_1_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_2"><b>case </b></a><a NAME="E1:50_1_1_1_1_2"/>
<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:50_1_1_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:50_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">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> 
 ;<br/><b>hence </b><a NAME="E4:50_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:50_1_2"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c4">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> 
 <b>and </b><br/><a NAME="E2:50_1_2"/><i><font color="Green" title="E50">A8</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font>;<br/><b>set </b><font color="Maroon" title="c8">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><b>set </b><font color="Maroon" title="c9">f2</font> = <font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font>;<br/><a NAME="E3:50_1_2"/><i><font color="Green" title="E51">A9</font></i>: 
 <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#D4" title="FINSEQ_5:def.4">FINSEQ_5:def 4</a></i>;<br/><a NAME="E4:50_1_2"/><i><font color="Green" title="E52">A10</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="#E1:50_1_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E5:50_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:50_1_2"/><b>then </b><i><font color="Green" title="E53">A11</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E5:50"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:50_1_2"/><i><font color="Green" title="E54">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></i>;<br/><a NAME="E8:50_1_2"/><i><font color="Green" title="E55">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/><a NAME="E9:50_1_2"/><b>then </b><i><font color="Green" title="E56">A14</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/><a NAME="E10:50_1_2"/><i><font color="Green" title="E57">A15</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</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> )
 <b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E11:50_1_2"/><i><font color="Green" title="E58">A16</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E12:50_1_2"/><b>then </b><i><font color="Green" title="E59">A17</font></i>: 
<font color="Maroon" title="c3">n</font> <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>
 <b>by </b><i><a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:50_1_2"/><i><font color="Green" title="E60">A18</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="txt" href="#E11:50_1_2"><i><font color="Green" title="E58">A16</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:50_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E5:50"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E15:50_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E16:50_1_2"/><b>then </b><i><font color="Green" title="E61">A19</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E2:50_1_2"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="#E7:50_1_2"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:50_1_2"/><b>then </b><i><font color="Green" title="E62">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="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>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E18:50_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E19:50_1_2"/><b>then </b><i><font color="Green" title="E63">A21</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="#E17:50_1_2"><i><font color="Green" title="E62">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_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="E1:50_1_2_1_1"/>
( <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_1"/><i><font color="Green" title="E64">A22</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_1"/>
<font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:50_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="#E6:50_1_2"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:50_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E1:50_1_2_1_1_1"><i><font color="Green" title="E64">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:50_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E65">A23</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><i><font color="Green" title="E66">A24</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E11:50_1_2"><i><font color="Green" title="E58">A16</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self" title="SPPOL_2:th.6">Th6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E5:50_1_2_1_1_1"><i><font color="Green" title="E65">A23</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self" title="SPPOL_2:th.6">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E5:50_1_2_1_1_1"><i><font color="Green" title="E65">A23</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self" title="SPPOL_2:th.3">Th3</a></i>
;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E11:50_1_2"><i><font color="Green" title="E58">A16</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_1"><i><font color="Green" title="E64">A22</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self" title="SPPOL_2:th.6">Th6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_1"><i><font color="Green" title="E64">A22</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self" title="SPPOL_2:th.6">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_1"><i><font color="Green" title="E64">A22</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self" title="SPPOL_2:th.3">Th3</a></i>
;<br/><b>hence </b><a NAME="E8:50_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E6:50_1_2_1_1_1"><i><font color="Green" title="E66">A24</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2"/>
<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E64">A25</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_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="E1:50_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E65">A26</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E66">A27</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:50_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E67">A28</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:50_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E68">A29</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><a NAME="E5:50_1_2_1_1_2_1_1_1"/>
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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E6:50_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:50_1_2"><i><font color="Green" title="E59">A17</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/><a NAME="E7:50_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:50_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E69">A30</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>;<br/><a NAME="E9:50_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E70">A31</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E9:50_1_2"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="#E2:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E66">A27</font></i></a>, <a class="txt" href="#E4:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E68">A29</font></i></a>, <a class="ref" href="topreal1.html#T11" title="TOPREAL1:th.11">TOPREAL1:11</a></i>;<br/><a NAME="E10:50_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E71">A32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E3:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E67">A28</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><i><font color="Green" title="E72">A33</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c3">n</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self" title="SPPOL_2:th.6">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E9:50_1_2"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E10:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E71">A32</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
;<br/><a NAME="E12:50_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E73">A34</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</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</span>)</span> <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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></i>;<br/><a NAME="E13:50_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E74">A35</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E14:50_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E75">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:50_1_2"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E15:50_1_2_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E16:50_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E17:50_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E76">A37</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E9:50_1_2"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A30</font></i></a>, <a class="txt" href="#E12:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E73">A34</font></i></a>, <a class="txt" href="#E13:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E74">A35</font></i></a>, <a class="txt" href="#E14:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E75">A36</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E18:50_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E77">A38</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">p</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#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E2:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E66">A27</font></i></a>, <a class="txt" href="#E4:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E68">A29</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A30</font></i></a>, <a class="ref" href="topreal1.html#T14" title="TOPREAL1:th.14">TOPREAL1:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E78">A39</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><i><font color="Green" title="E79">A40</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E11:50_1_2"><i><font color="Green" title="E58">A16</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self" title="SPPOL_2:th.6">Th6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E2:50_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green" title="E78">A39</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self" title="SPPOL_2:th.6">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E2:50_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green" title="E78">A39</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self" title="SPPOL_2:th.3">Th3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1"/>
( <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E80">A41</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E81">A42</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/><a NAME="E3:50_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E66">A27</font></i></a>, <a class="txt" href="#E4:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E68">A29</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A30</font></i></a>, <a class="txt" href="#E9:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A31</font></i></a>, <a class="txt" href="#E11:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green" title="E79">A40</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E80">A41</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E4:50_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E81">A42</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:50_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">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> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E6:50_1_2_1_1_2_1_1_1_2_1_1_2_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2"><i><font color="Green" title="E64">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E80">A43</font></i>: 
<font color="Maroon" title="c3">n</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="c5">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1"/>
( <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"/><i><font color="Green" title="E81">A44</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E82">A45</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><b>then </b><i><font color="Green" title="E83">A46</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self" title="SPPOL_2:th.6">Th6</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E82">A45</font></i></a>, <a class="ref" href="sppol_2.html#T6" target="_self" title="SPPOL_2:th.6">Th6</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E82">A45</font></i></a>, <a class="ref" href="sppol_2.html#T3" target="_self" title="SPPOL_2:th.3">Th3</a></i>
;<br/><a NAME="E3:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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>
 <b>by </b><i><a class="txt" href="#E16:50_1_2"><i><font color="Green" title="E61">A19</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E4:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E84">A47</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="#E4:50_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E82">A45</font></i></a>, <a class="txt" href="#E2:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E83">A46</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/><b>assume </b><a NAME="E5:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/>
not  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E85">A48</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></i>;<br/><a NAME="E8:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E86">A49</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/><a NAME="E9:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E87">A50</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="#E7:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E85">A48</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E88">A51</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E7:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E85">A48</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E11:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E9:50_1_2"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="#E2:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E66">A27</font></i></a>, <a class="txt" href="#E4:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E68">A29</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A30</font></i></a>, <a class="txt" href="#E9:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A31</font></i></a>, <a class="txt" href="#E17:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E76">A37</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"><i><font color="Green" title="E81">A44</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E12:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E84">A47</font></i></a>, <a class="txt" href="#E9:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E87">A50</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E13:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E89">A52</font></i>: 
<font color="Maroon" title="c10">x</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">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E14:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E4:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E68">A29</font></i></a>, <a class="txt" href="#E11:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E72">A33</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E15:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c3">n</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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"><i><font color="Green" title="E81">A44</font></i></a>, <a class="txt" href="#E10:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E88">A51</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E16:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">p</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">n</font>
 <b>by </b><i><a class="txt" href="#E11:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="#E17:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E76">A37</font></i></a>, <a class="txt" href="#E18:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E77">A38</font></i></a>, <a class="txt" href="#E13:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E89">A52</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E17:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:50"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_1"><i><font color="Green" title="E86">A49</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green" title="E78">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E82">A53</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/><a NAME="E4:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:50_1_2"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="#E2:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E66">A27</font></i></a>, <a class="txt" href="#E4:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E68">A29</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A30</font></i></a>, <a class="txt" href="#E9:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A31</font></i></a>, <a class="txt" href="#E17:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E76">A37</font></i></a>, <a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green" title="E79">A40</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"><i><font color="Green" title="E81">A44</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E5:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"><i><font color="Green" title="E82">A53</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E6:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">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> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E7:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E81">A54</font></i>: 
<font color="Maroon" title="c3">n</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="c5">j</font>
 <b>by </b><i><a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2"><i><font color="Green" title="E80">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E82">A55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">j'</font> = <font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E4:50_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="#E6:50_1_2"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">nj</font> = <font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2"><i><font color="Green" title="E80">A43</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E6:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><i><font color="Green" title="E83">A56</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">nj</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E7:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><i><font color="Green" title="E84">A57</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">nj</font>
 <b>by </b><i><a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green" title="E82">A55</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E8:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><i><font color="Green" title="E85">A58</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">nj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j'</font>
 ;<br/><a NAME="E9:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green" title="E78">A39</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green" title="E81">A54</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E86">A59</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j'</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c11">nj</font>
<b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E6:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green" title="E83">A56</font></i></a>, <a class="txt" href="#E7:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green" title="E84">A57</font></i></a>, <a class="ref" href="sppol_2.html#T7" target="_self" title="SPPOL_2:th.7">Th7</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c10">j'</font>
<b>by </b><i><a class="txt" href="#E12:50_1_2"><i><font color="Green" title="E59">A17</font></i></a>, <a class="txt" href="#E7:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green" title="E84">A57</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green" title="E85">A58</font></i></a>, <a class="ref" href="sppol_2.html#T4" target="_self" title="SPPOL_2:th.4">Th4</a></i>
;<br/><b>hence </b><a NAME="E13:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_1"><i><font color="Green" title="E79">A40</font></i></a>, <a class="txt" href="#E11:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2_1_1_2"><i><font color="Green" title="E86">A59</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:50_1_2_1_1_2_1_1_1_2_1_1_2_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:50_1_2_1_1_2_1_1_1_2_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E78">A60</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E79">A61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E5:50"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">j'</font> = <font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E4:50_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="#E6:50_1_2"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">nj</font> = <font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E5:50"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E78">A60</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E5:50_1_2_1_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E80">A62</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">nj</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E6:50_1_2_1_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E81">A63</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">nj</font>
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E79">A61</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E7:50_1_2_1_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E82">A64</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">nj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j'</font>
 ;<br/><i><font color="Green" title="E83">A65</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c11">nj</font>
<b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E5:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E80">A62</font></i></a>, <a class="txt" href="#E6:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E81">A63</font></i></a>, <a class="ref" href="sppol_2.html#T7" target="_self" title="SPPOL_2:th.7">Th7</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c10">j'</font>
<b>by </b><i><a class="txt" href="#E12:50_1_2"><i><font color="Green" title="E59">A17</font></i></a>, <a class="txt" href="#E6:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E81">A63</font></i></a>, <a class="txt" href="#E7:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E82">A64</font></i></a>, <a class="ref" href="sppol_2.html#T4" target="_self" title="SPPOL_2:th.4">Th4</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_2_2_1"/>
( <font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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 or <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E79">A61</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:50_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j'</font>
 <b>by </b><i><a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E78">A60</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E4:50_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E84">A66</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c10">j'</font>
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/><a NAME="E5:50_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c10">j'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E66">A27</font></i></a>, <a class="txt" href="#E4:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E68">A29</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A30</font></i></a>, <a class="txt" href="#E9:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A31</font></i></a>, <a class="txt" href="#E11:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E78">A60</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E83">A65</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E6:50_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E4:50_1_2_1_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E84">A66</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E7:50_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">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> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E8:50_1_2_1_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green" title="E84">A67</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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>
 <b>by </b><i><a class="txt" href="#E2:50_1_2"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="#E7:50_1_2"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E85">A68</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</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#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c10">j'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">j'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green" title="E84">A67</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/><b>assume </b><a NAME="E4:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/>
not  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green" title="E86">A69</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></i>;<br/><a NAME="E7:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green" title="E87">A70</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="#E10:50_1_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/><a NAME="E8:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green" title="E88">A71</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E6:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green" title="E86">A69</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E9:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green" title="E89">A72</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="#E6:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green" title="E86">A69</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E66">A27</font></i></a>, <a class="txt" href="#E4:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E68">A29</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A30</font></i></a>, <a class="txt" href="#E9:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A31</font></i></a>, <a class="txt" href="#E11:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E78">A60</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E8:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E83">A65</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green" title="E84">A67</font></i></a>, <a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green" title="E85">A68</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green" title="E88">A71</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E12:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E90">A73</font></i>: 
<font color="Maroon" title="c12">x</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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E13:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E8:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A30</font></i></a>, <a class="txt" href="#E17:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E76">A37</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E14:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c3">n</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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:50_1_2_1_1_2_1_1_1_2_1_2"><i><font color="Green" title="E78">A60</font></i></a>, <a class="txt" href="#E9:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green" title="E89">A72</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E15:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">p</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"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E11:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="#E17:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E76">A37</font></i></a>, <a class="txt" href="#E18:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E77">A38</font></i></a>, <a class="txt" href="#E12:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green" title="E90">A73</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E16:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:50"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="#E7:50_1_2_1_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green" title="E87">A70</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:50_1_2_1_1_2_1_1_1_2_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:50_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b><i><font color="Green" title="E78">A74</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E1:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E65">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">j'</font> = <font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E4:50_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="#E6:50_1_2"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:50_1_2_1_1_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c3">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">nj</font> = <font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E5:50"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green" title="E78">A74</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:50_1_2_1_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E79">A75</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">nj</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E8:50_1_2_1_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E80">A76</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">nj</font>
 <b>by </b><i><a class="txt" href="#E5:50"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green" title="E78">A74</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E9:50_1_2_1_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E81">A77</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">nj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j'</font>
 ;<br/><i><font color="Green" title="E82">A78</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c11">nj</font>
<b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E7:50_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green" title="E79">A75</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green" title="E80">A76</font></i></a>, <a class="ref" href="sppol_2.html#T7" target="_self" title="SPPOL_2:th.7">Th7</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c10">j'</font>
<b>by </b><i><a class="txt" href="#E12:50_1_2"><i><font color="Green" title="E59">A17</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green" title="E80">A76</font></i></a>, <a class="txt" href="#E9:50_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green" title="E81">A77</font></i></a>, <a class="ref" href="sppol_2.html#T4" target="_self" title="SPPOL_2:th.4">Th4</a></i>
;<br/><a NAME="E11:50_1_2_1_1_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c3">n</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="c10">j'</font>
 <b>by </b><i><a class="txt" href="#E5:50"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green" title="E78">A74</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E12:50_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b><i><font color="Green" title="E83">A79</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c10">j'</font>
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/><a NAME="E13:50_1_2_1_1_2_1_1_1_2_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</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#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c10">j'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:50_1_2"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="#E2:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E66">A27</font></i></a>, <a class="txt" href="#E4:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E68">A29</font></i></a>, <a class="txt" href="#E8:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A30</font></i></a>, <a class="txt" href="#E9:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A31</font></i></a>, <a class="txt" href="#E17:50_1_2_1_1_2_1_1_1"><i><font color="Green" title="E76">A37</font></i></a>, <a class="txt" href="#E10:50_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green" title="E82">A78</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E14:50_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E12:50_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green" title="E83">A79</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E15:50_1_2_1_1_2_1_1_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">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> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E16:50_1_2_1_1_2_1_1_1_2_1_3"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E3:50_1_2_1_1_2_1_1_1_2_1_3"><i><font color="Green" title="E78">A74</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E20:50_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E65">A80</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E66">A81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">ni</font> = <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E65">A80</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E4:50_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E67">A82</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ni</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">i'</font> = <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:50_1_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E6:50_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E68">A83</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">ni</font>
 <b>by </b><i><a class="txt" href="#E2:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E66">A81</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E7:50_1_2_1_1_2_1_1_2"/>
 <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#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_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E8:50_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E19:50_1_2"><i><font color="Green" title="E63">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:50_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E10:50_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E69">A84</font></i>: 
<font color="Maroon" title="c10">ni</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> <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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">n</font>
 ;<br/><a NAME="E11:50_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E70">A85</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">ni</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i'</font>
 ;<br/><i><font color="Green" title="E71">A86</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c10">ni</font>
<b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E4:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E67">A82</font></i></a>, <a class="txt" href="#E6:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A83</font></i></a>, <a class="ref" href="sppol_2.html#T7" target="_self" title="SPPOL_2:th.7">Th7</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c11">i'</font>
<b>by </b><i><a class="txt" href="#E6:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A83</font></i></a>, <a class="txt" href="#E10:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E69">A84</font></i></a>, <a class="txt" href="#E11:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A85</font></i></a>, <a class="ref" href="sppol_2.html#T5" target="_self" title="SPPOL_2:th.5">Th5</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c12">j'</font> = <font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E4:50_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="#E6:50_1_2"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:50_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E72">A87</font></i>: 
<font color="Maroon" title="c3">n</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="c5">j</font>
 <b>by </b><i><a class="txt" href="#E6:50_1_2"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E65">A80</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">nj</font> = <font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E6:50_1_2"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="#E1:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E65">A80</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E16:50_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E73">A88</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">nj</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E17:50_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E14:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E72">A87</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E18:50_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E74">A89</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">nj</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E19:50_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E75">A90</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">nj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j'</font>
 ;<br/><a NAME="E20:50_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j'</font>
 <b>by </b><i><a class="txt" href="#E5:50"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E21:50_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E76">A91</font></i>: 
<font color="Maroon" title="c11">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="c12">j'</font>
 ;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font> = 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c13">nj</font>
<b>by </b><i><a class="txt" href="#E3:50_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E8:50_1_2"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="#E13:50_1_2"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="#E16:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A88</font></i></a>, <a class="txt" href="#E18:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E74">A89</font></i></a>, <a class="ref" href="sppol_2.html#T7" target="_self" title="SPPOL_2:th.7">Th7</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c12">j'</font>
<b>by </b><i><a class="txt" href="#E12:50_1_2"><i><font color="Green" title="E59">A17</font></i></a>, <a class="txt" href="#E18:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E74">A89</font></i></a>, <a class="txt" href="#E19:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E75">A90</font></i></a>, <a class="ref" href="sppol_2.html#T4" target="_self" title="SPPOL_2:th.4">Th4</a></i>
;<br/><b>hence </b><a NAME="E23:50_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E12:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A86</font></i></a>, <a class="txt" href="#E21:50_1_2_1_1_2_1_1_2"><i><font color="Green" title="E76">A91</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:50_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E21:50_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
