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<span class="kw">let </span><font color="Maroon" title="c1">e</font>, <font color="Maroon" title="c2">f</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/> for <font color="Olive" title="b3">S</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>  st (  for <font color="Olive" title="b4">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b3">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">i</font> is  <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>  &amp; <font color="Olive" title="b3">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b3">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">B</font> ) ) holds <br/> ex <font color="Olive" title="b4">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  not <font color="Olive" title="b4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> &amp; (  for <font color="Olive" title="b5">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b6">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">k</font> &amp; <font color="Olive" title="b4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b6">k</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies  for <font color="Olive" title="b1">S</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>  st (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b1">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">i</font> is  <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>  &amp; <font color="Olive" title="b1">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font> ) ) holds <br/> ex <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b4">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">k</font> ) ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:15"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>  st (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b1">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">i</font> is  <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>  &amp; <font color="Olive" title="b1">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font> ) ) holds <br/> ex <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b4">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">k</font> ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">S</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> is  <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>  &amp; <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font> ) ) implies  ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  not <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">k</font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:15"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> is  <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>  &amp; <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  not <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">k</font> ) ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>] <span class="kw">means </span>(  not $2 is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; $2 is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  &amp; $2 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font> &amp; $2 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font> &amp; $2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E5:15"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">u</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">u</font>]</span><br/>

<a NAME="E1:15_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font> is  <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a></span>;<br/>
<a NAME="E2:15_1"/>
<font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:15_1"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E5:15_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<a NAME="E6:15_1"/>
<font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:15_1"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E9:15_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_1_1"/>
( <font color="Maroon" title="c10">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">y</font> or <font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_1"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">u</font>]</span><br/><div class="add"><span class="kw">take </span>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font>,<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>]</span><br/><span class="kw">thus </span><a NAME="E2:15_1_1_1"/>
(  not <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="measure5.html#T14" onmouseover="rs('measure5/T14')" onmouseout="rh()">MEASURE5:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><a NAME="E3:15_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:15_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><a NAME="E5:15_1_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:15_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><span class="kw">thus </span><a NAME="E7:15_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_2"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">u</font>]</span><br/><div class="add"><span class="kw">take </span>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font>,<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>]</span><br/><span class="kw">thus </span><a NAME="E2:15_1_1_2"/>
(  not <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="measure5.html#T14" onmouseover="rs('measure5/T14')" onmouseout="rh()">MEASURE5:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><a NAME="E3:15_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:15_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><a NAME="E5:15_1_1_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:15_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><span class="kw">thus </span><a NAME="E7:15_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c6">T</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:15"/><span class="lab"><font color="Green" title="E17">A13</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>);<br/></span>
<a NAME="E8:15"/>
<font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/>
<a NAME="E9:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A14</font></span>: 
 not <font color="Maroon" title="c4">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">SA</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:15"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c8">SB</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E13:15"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>,  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>] <span class="kw">means </span>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 &amp; $3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 &amp;  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default">$2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $3</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> );<br/>
<a NAME="E14:15"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">ai</font>, <font color="Olive" title="b3">bi</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">ai</font>,<font color="Olive" title="b3">bi</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">ai</font>, <font color="Olive" title="b2">bi</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Olive" title="b1">ai</font>,<font color="Olive" title="b2">bi</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">Si</font> = <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">i</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/>
<a NAME="E2:15_2"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
<font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/>
<a NAME="E3:15_2"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
<font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Si</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E19">A15</font></span></a></span>;<br/>
<a NAME="E4:15_2"/><span class="lab"><font color="Green" title="E24">A20</font></span>: 
<font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Si</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E20">A16</font></span></a></span>;<br/>
<a NAME="E5:15_2"/>
<font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">SAi</font> = <font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c12">SBi</font> = <font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">i</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E3:15_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="#E4:15_2"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="ref" href="jct_misc.html#T5" target="_self" onmouseover="rs('jct_misc/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">ai</font>, <font color="Maroon" title="c14">bi</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E8:15_2"/><span class="lab"><font color="Green" title="E25">A21</font></span>: 
<font color="Maroon" title="c13">ai</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">SAi</font>
 <span class="kw">and </span><br/><a NAME="E9:15_2"/><span class="lab"><font color="Green" title="E26">A22</font></span>: 
<font color="Maroon" title="c14">bi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">SBi</font>
 <span class="kw">and </span><br/><a NAME="E10:15_2"/><span class="lab"><font color="Green" title="E27">A23</font></span>: 
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c11">SAi</font>,<font color="Maroon" title="c12">SBi</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ai</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">bi</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jct_misc.html#T9" target="_self" onmouseover="rs('jct_misc/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">ai</font> = <font color="Maroon" title="c13">ai</font>, <font color="Maroon" title="c16">bi</font> = <font color="Maroon" title="c14">bi</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">ai</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">bi</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c15">ai</font>,<font color="Olive" title="b1">bi</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c16">bi</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c15">ai</font>,<font color="Maroon" title="c16">bi</font>]</span><br/>

<span class="kw">thus </span><a NAME="E12:15_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c15">ai</font>,<font color="Maroon" title="c16">bi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_2"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="txt" href="#E9:15_2"><span class="lab"><font color="Green" title="E26">A22</font></span></a>, <a class="txt" href="#E10:15_2"><span class="lab"><font color="Green" title="E27">A23</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c9">sa</font>, <font color="Maroon" title="c10">sb</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E16:15"/><span class="lab"><font color="Green" title="E22">A24</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="jct_misc.html#S2" target="_self" onmouseover="rs('jct_misc/S2')" onmouseout="rh()">JCT_MISC:sch 2</a>(<a class="txt" href="#E14:15"><span class="lab"><font color="Green" title="E21">A17</font></span></a>);<br/></span>
<a NAME="E17:15"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">sa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">sa</font> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_3"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">sa</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:15_3"/><span class="lab"><font color="Green" title="E23">A25</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">sa</font>
 <span class="kw">and </span><br/><a NAME="E4:15_3"/><span class="lab"><font color="Green" title="E24">A26</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">sa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">n</font> = <font color="Maroon" title="c12">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_3"><span class="lab"><font color="Green" title="E23">A25</font></span></a></span>;<br/>
<a NAME="E6:15_3"/>
<font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/>
<a NAME="E7:15_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E4:15_3"><span class="lab"><font color="Green" title="E24">A26</font></span></a></span>;<br/>
<a NAME="E8:15_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:15_3"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">ga</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E19:15"/><span class="lab"><font color="Green" title="E23">A27</font></span>: 
<font color="Maroon" title="c11">ga</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c9">sa</font>
 <span class="kw">and </span><br/><a NAME="E20:15"/><span class="lab"><font color="Green" title="E24">A28</font></span>: 
<font color="Maroon" title="c11">ga</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E21:15"/><span class="lab"><font color="Green" title="E25">A29</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c11">ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D3" onmouseover="rs('rcomp_1/D3')" onmouseout="rh()">RCOMP_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">Nseq</font> being   <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a><span class="kw"> such that </span><br/><a NAME="E23:15"/><span class="lab"><font color="Green" title="E26">A30</font></span>: 
<font color="Maroon" title="c11">ga</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">sa</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15"><span class="lab"><font color="Green" title="E23">A27</font></span></a>, <a class="ref" href="valued_0.html#D17" onmouseover="rs('valued_0/D17')" onmouseout="rh()">VALUED_0:def 17</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">gb</font> = <font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font>;<br/>
<a NAME="E24:15"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font></span>)</span> or <font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_4"/>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:15_4"/><span class="lab"><font color="Green" title="E27">A31</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:15_4"/><span class="lab"><font color="Green" title="E28">A32</font></span>: 
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">n</font> = <font color="Maroon" title="c15">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_4"><span class="lab"><font color="Green" title="E27">A31</font></span></a></span>;<br/>
<a NAME="E6:15_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>;<br/>
<a NAME="E7:15_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/>
<a NAME="E8:15_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E4:15_4"><span class="lab"><font color="Green" title="E28">A32</font></span></a></span>;<br/>
<a NAME="E9:15_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:15_4"/>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">fb</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E26:15"/><span class="lab"><font color="Green" title="E27">A33</font></span>: 
<font color="Maroon" title="c14">fb</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font>
 <span class="kw">and </span><br/><a NAME="E27:15"/><span class="lab"><font color="Green" title="E28">A34</font></span>: 
<font color="Maroon" title="c14">fb</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E28:15"/><span class="lab"><font color="Green" title="E29">A35</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D3" onmouseover="rs('rcomp_1/D3')" onmouseout="rh()">RCOMP_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">Nseq1</font> being   <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a><span class="kw"> such that </span><br/><a NAME="E30:15"/><span class="lab"><font color="Green" title="E30">A36</font></span>: 
<font color="Maroon" title="c14">fb</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:15"><span class="lab"><font color="Green" title="E27">A33</font></span></a>, <a class="ref" href="valued_0.html#D17" onmouseover="rs('valued_0/D17')" onmouseout="rh()">VALUED_0:def 17</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">fa</font> = <font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">r</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">d0</font> =  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c19">ff</font> = <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span>;<br/>
<a NAME="E31:15"/><span class="lab"><font color="Green" title="E31">A37</font></span>: 
<font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:15"><span class="lab"><font color="Green" title="E24">A28</font></span></a>, <a class="ref" href="seq_4.html#T16" onmouseover="rs('seq_4/T16')" onmouseout="rh()">SEQ_4:16</a></span>;<br/>
<a NAME="E32:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A38</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:15"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="seq_2.html#T5" onmouseover="rs('seq_2/T5')" onmouseout="rh()">SEQ_2:5</a></span>;<br/>
<a NAME="E33:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A39</font></span>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T7" onmouseover="rs('seq_2/T7')" onmouseout="rh()">SEQ_2:7</a></span>;<br/>
<a NAME="E34:15"/>
<font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/>
<a NAME="E35:15"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
;<br/>
<a NAME="E36:15"/><span class="kw">then </span>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="ref" href="jct_misc.html#T11" target="_self" onmouseover="rs('jct_misc/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<a NAME="E37:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A40</font></span>: 
<span class="p1">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:15"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="txt" href="#E31:15"><span class="lab"><font color="Green" title="E31">A37</font></span></a>, <a class="ref" href="seq_2.html#T6" onmouseover="rs('seq_2/T6')" onmouseout="rh()">SEQ_2:6</a></span>
<br/>.= 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:15"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="seq_2.html#T8" onmouseover="rs('seq_2/T8')" onmouseout="rh()">SEQ_2:8</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">k0</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E40:15"/><span class="lab"><font color="Green" title="E35">A41</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c20">k0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:15"><span class="lab"><font color="Green" title="E33">A39</font></span></a>, <a class="txt" href="#E37:15"><span class="lab"><font color="Green" title="E34">A40</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>;<br/>
<a NAME="E41:15"/><span class="lab"><font color="Green" title="E36">A42</font></span>: 
<font color="Maroon" title="c20">k0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">k</font> ) ) )</span><br/>

<a NAME="E42:15"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c11">ga</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:15"><span class="lab"><font color="Green" title="E24">A28</font></span></a>, <a class="ref" href="seq_4.html#T17" onmouseover="rs('seq_4/T17')" onmouseout="rh()">SEQ_4:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E43:15"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:15"><span class="lab"><font color="Green" title="E25">A29</font></span></a>, <a class="txt" href="#E28:15"><span class="lab"><font color="Green" title="E29">A35</font></span></a>, <a class="ref" href="jct_misc.html#T1" target="_self" onmouseover="rs('jct_misc/T1')" onmouseout="rh()">Th1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">k</font> ) ) )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_6"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c21">i</font> = <font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">di</font> =  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>);<br/><span class="lab"><font color="Green" title="E37">A43</font></span>: 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default">2</span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>

;<br/><span class="lab"><font color="Green" title="E38">A44</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font> = 
<font color="Maroon" title="c11">ga</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a>, <a class="txt" href="#E41:15"><span class="lab"><font color="Green" title="E36">A42</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E26">A30</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/><a NAME="E3:15_6"/>
<font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/><a NAME="E4:15_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E5:15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A45</font></span>: 
 not <font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E20">A16</font></span></a></span>;<br/><a NAME="E6:15_6"/><span class="lab"><font color="Green" title="E40">A46</font></span>: 
<font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E20">A16</font></span></a></span>;<br/><a NAME="E7:15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A47</font></span>: 
<font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/><a NAME="E8:15_6"/><span class="lab"><font color="Green" title="E42">A48</font></span>: 
<font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_6"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/><a NAME="E9:15_6"/><span class="lab"><font color="Green" title="E43">A49</font></span>: 
<font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E19">A15</font></span></a></span>;<br/><a NAME="E10:15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A50</font></span>: 
<font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/><a NAME="E11:15_6"/>
<font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/><a NAME="E12:15_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E13:15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A51</font></span>: 
 not <font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E19">A15</font></span></a></span>;<br/><a NAME="E14:15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A52</font></span>: 
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_6"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="txt" href="#E10:15_6"><span class="lab"><font color="Green" title="E44">A50</font></span></a>, <a class="txt" href="#E7:15_6"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="ref" href="jct_misc.html#T8" target="_self" onmouseover="rs('jct_misc/T8')" onmouseout="rh()">Th8</a></span>;<br/><a NAME="E15:15_6"/>
<span class="p1">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15_6"><span class="lab"><font color="Green" title="E45">A51</font></span></a>, <a class="txt" href="#E5:15_6"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="txt" href="#E10:15_6"><span class="lab"><font color="Green" title="E44">A50</font></span></a>, <a class="txt" href="#E7:15_6"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="ref" href="jct_misc.html#T8" target="_self" onmouseover="rs('jct_misc/T8')" onmouseout="rh()">Th8</a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><a NAME="E16:15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A53</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</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="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T9" onmouseover="rs('seq_1/T9')" onmouseout="rh()">SEQ_1:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" onmouseover="rs('seq_1/T7')" onmouseout="rh()">SEQ_1:7</a></span>
;<br/><a NAME="E18:15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A54</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:15"><span class="lab"><font color="Green" title="E35">A41</font></span></a></span>;<br/><a NAME="E19:15_6"/>
(  not <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/><a NAME="E20:15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A55</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#T14" onmouseover="rs('measure5/T14')" onmouseout="rh()">MEASURE5:14</a></span>;<br/><a NAME="E21:15_6"/><span class="lab"><font color="Green" title="E50">A56</font></span>: 
<font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><a NAME="E22:15_6"/><span class="lab"><font color="Green" title="E51">A57</font></span>: 
<font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15_6"><span class="lab"><font color="Green" title="E43">A49</font></span></a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/><span class="lab"><font color="Green" title="E52">A58</font></span>: <font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:15"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a>, <a class="txt" href="#E41:15"><span class="lab"><font color="Green" title="E36">A42</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/><a NAME="E24:15_6"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)
 ;<br/><a NAME="E25:15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A59</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:15_6"><span class="lab"><font color="Green" title="E48">A54</font></span></a>, <a class="txt" href="#E2:15_6"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="txt" href="#E23:15_6"><span class="lab"><font color="Green" title="E52">A58</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><a NAME="E26:15_6"/><span class="lab"><font color="Green" title="E54">A60</font></span>: 
<font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><a NAME="E27:15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A61</font></span>: 
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:15_6"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="ref" href="jct_misc.html#T7" target="_self" onmouseover="rs('jct_misc/T7')" onmouseout="rh()">Th7</a></span>;<br/><div><span class="lab"><font color="Green" title="E56">A62</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_6_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_6_5_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_6_5_1"/>
( <font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> or <font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_6_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_6_5_1_1"/>
<font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:15_6_5_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><a NAME="E3:15_6_5_1_1"/><span class="kw">then </span>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:15_6"><span class="lab"><font color="Green" title="E55">A61</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><a NAME="E4:15_6_5_1_1"/><span class="kw">then </span>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:15_6"><span class="lab"><font color="Green" title="E46">A52</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:15_6_5_1_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:15_6"><span class="lab"><font color="Green" title="E53">A59</font></span></a>, <a class="txt" href="#E1:15_6"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:15_6_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A63</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T2" onmouseover="rs('rcomp_1/T2')" onmouseout="rh()">RCOMP_1:2</a></span>;<br/><a NAME="E7:15_6_5_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:15_6"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E26:15_6"><span class="lab"><font color="Green" title="E54">A60</font></span></a>, <a class="txt" href="#E21:15_6"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="txt" href="#E22:15_6"><span class="lab"><font color="Green" title="E51">A57</font></span></a>, <a class="txt" href="#E8:15_6"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="ref" href="xxreal_2.html#D12" onmouseover="rs('xxreal_2/D12')" onmouseout="rh()">XXREAL_2:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E8:15_6_5_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_6_5_1_1"><span class="lab"><font color="Green" title="E57">A63</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_6_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_6_5_1_2"/><span class="lab"><font color="Green" title="E57">A64</font></span>: 
<font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:15_6_5_1_2"/><span class="lab"><font color="Green" title="E58">A65</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uniform1.html#T11" onmouseover="rs('uniform1/T11')" onmouseout="rh()">UNIFORM1:11</a></span>;<br/><a NAME="E3:15_6_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_6_5_1_2"><span class="lab"><font color="Green" title="E57">A64</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><a NAME="E4:15_6_5_1_2"/><span class="kw">then </span>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:15_6"><span class="lab"><font color="Green" title="E55">A61</font></span></a>, <a class="txt" href="#E2:15_6_5_1_2"><span class="lab"><font color="Green" title="E58">A65</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:15_6_5_1_2"/><span class="kw">then </span>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:15_6"><span class="lab"><font color="Green" title="E46">A52</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:15_6_5_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:15_6"><span class="lab"><font color="Green" title="E53">A59</font></span></a>, <a class="txt" href="#E1:15_6"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:15_6_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A66</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T2" onmouseover="rs('rcomp_1/T2')" onmouseout="rh()">RCOMP_1:2</a></span>;<br/><a NAME="E8:15_6_5_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:15_6"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E26:15_6"><span class="lab"><font color="Green" title="E54">A60</font></span></a>, <a class="txt" href="#E21:15_6"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="txt" href="#E22:15_6"><span class="lab"><font color="Green" title="E51">A57</font></span></a>, <a class="txt" href="#E8:15_6"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="ref" href="xxreal_2.html#D12" onmouseover="rs('xxreal_2/D12')" onmouseout="rh()">XXREAL_2:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E9:15_6_5_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15_6_5_1_2"><span class="lab"><font color="Green" title="E59">A66</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E29:15_6"/><span class="lab"><font color="Green" title="E57">A67</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_6_6"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_6_6"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">B</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:15_6"><span class="lab"><font color="Green" title="E57">A67</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_6_6_1"><span class="kw">suppose </span></a><a NAME="E1:15_6_6_1"/><span class="lab"><font color="Green" title="E58">A68</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:15_6_6_1"/>
<font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E20">A16</font></span></a></span>;<br/><a NAME="E3:15_6_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A69</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:15_6"><span class="lab"><font color="Green" title="E56">A62</font></span></a>, <a class="txt" href="#E1:15_6_6_1"><span class="lab"><font color="Green" title="E58">A68</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E60">A70</font></span>: <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default">2</span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T67" onmouseover="rs('complex1/T67')" onmouseout="rh()">COMPLEX1:67</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
;<br/><a NAME="E5:15_6_6_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>
 ;<br/><a NAME="E6:15_6_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A71</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T56" onmouseover="rs('complex1/T56')" onmouseout="rh()">COMPLEX1:56</a></span>;<br/><a NAME="E7:15_6_6_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:15_6"><span class="lab"><font color="Green" title="E48">A54</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:15_6_6_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_6_6_1"><span class="lab"><font color="Green" title="E61">A71</font></span></a>, <a class="txt" href="#E4:15_6_6_1"><span class="lab"><font color="Green" title="E60">A70</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E9:15_6_6_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:15_6"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="txt" href="#E23:15_6"><span class="lab"><font color="Green" title="E52">A58</font></span></a></span>;<br/><a NAME="E10:15_6_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A72</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15_6"><span class="lab"><font color="Green" title="E47">A53</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E11:15_6_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:15_6"><span class="lab"><font color="Green" title="E38">A44</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E12:15_6_6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_6_6_1"><span class="lab"><font color="Green" title="E59">A69</font></span></a>, <a class="txt" href="#E10:15_6_6_1"><span class="lab"><font color="Green" title="E62">A72</font></span></a>, <a class="ref" href="jct_misc.html#T7" target="_self" onmouseover="rs('jct_misc/T7')" onmouseout="rh()">Th7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_6_6_2"><span class="kw">suppose </span></a><a NAME="E1:15_6_6_2"/><span class="lab"><font color="Green" title="E58">A73</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:15_6_6_2"/>
<font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E19">A15</font></span></a></span>;<br/><a NAME="E3:15_6_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A74</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:15_6"><span class="lab"><font color="Green" title="E56">A62</font></span></a>, <a class="txt" href="#E1:15_6_6_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E60">A75</font></span>: <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default">2</span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T67" onmouseover="rs('complex1/T67')" onmouseout="rh()">COMPLEX1:67</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
;<br/><a NAME="E5:15_6_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>
 ;<br/><a NAME="E6:15_6_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A76</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T56" onmouseover="rs('complex1/T56')" onmouseout="rh()">COMPLEX1:56</a></span>;<br/><span class="lab"><font color="Green" title="E62">A77</font></span>: <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="uniform1.html#T11" onmouseover="rs('uniform1/T11')" onmouseout="rh()">UNIFORM1:11</a></span>
<br/>.= 
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:15_6"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="txt" href="#E23:15_6"><span class="lab"><font color="Green" title="E52">A58</font></span></a></span>
;<br/><a NAME="E8:15_6_6_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:15_6"><span class="lab"><font color="Green" title="E48">A54</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E9:15_6_6_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_6_6_2"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E4:15_6_6_2"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E10:15_6_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A78</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15_6"><span class="lab"><font color="Green" title="E47">A53</font></span></a>, <a class="txt" href="#E7:15_6_6_2"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E11:15_6_6_2"/>
<font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">k0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E23:15_6"><span class="lab"><font color="Green" title="E52">A58</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E12:15_6_6_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_6_6_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E10:15_6_6_2"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="jct_misc.html#T7" target="_self" onmouseover="rs('jct_misc/T7')" onmouseout="rh()">Th7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E45:15"/>
 not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">k</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c21">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c22">k</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>);<br/>
<a NAME="E46:15"/><span class="lab"><font color="Green" title="E37">A79</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c23">j</font> = <font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> )</span><br/>

<span class="lab"><font color="Green" title="E38">A80</font></span>: <font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c12">Nseq</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:15"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a>, <a class="txt" href="#E46:15"><span class="lab"><font color="Green" title="E37">A79</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<a NAME="E48:15"/><span class="lab"><font color="Green" title="E39">A81</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a></span>;<br/>
<a NAME="E49:15"/><span class="lab"><font color="Green" title="E40">A82</font></span>: 
<font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/>
<a NAME="E50:15"/>
<font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/>
<a NAME="E51:15"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E52:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A83</font></span>: 
 not <font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E20">A16</font></span></a></span>;<br/>
<a NAME="E53:15"/><span class="lab"><font color="Green" title="E42">A84</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">Nseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E54:15"/>
<font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/>
<a NAME="E55:15"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E56:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A85</font></span>: 
 not <font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E19">A15</font></span></a></span>;<br/>
<a NAME="E57:15"/><span class="lab"><font color="Green" title="E44">A86</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#T14" onmouseover="rs('seqm_3/T14')" onmouseout="rh()">SEQM_3:14</a></span>;<br/>
<a NAME="E58:15"/><span class="lab"><font color="Green" title="E45">A87</font></span>: 
<font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E19">A15</font></span></a></span>;<br/>
<a NAME="E59:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A88</font></span>: 
<font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:15"><span class="lab"><font color="Green" title="E37">A79</font></span></a>, <a class="ref" href="seq_1.html#T9" onmouseover="rs('seq_1/T9')" onmouseout="rh()">SEQ_1:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">fb</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:15"><span class="lab"><font color="Green" title="E37">A79</font></span></a>, <a class="ref" href="seq_1.html#T7" onmouseover="rs('seq_1/T7')" onmouseout="rh()">SEQ_1:7</a></span>
;<br/>
<a NAME="E61:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A89</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p0">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c14">fb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p0">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:15"><span class="lab"><font color="Green" title="E35">A41</font></span></a>, <a class="txt" href="#E46:15"><span class="lab"><font color="Green" title="E37">A79</font></span></a>, <a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a></span>;<br/>
<a NAME="E62:15"/><span class="lab"><font color="Green" title="E48">A90</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)
 
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span> = 
<font color="Maroon" title="c11">ga</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a>, <a class="txt" href="#E46:15"><span class="lab"><font color="Green" title="E37">A79</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E26">A30</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<a NAME="E64:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A91</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:15"><span class="lab"><font color="Green" title="E47">A89</font></span></a>, <a class="txt" href="#E47:15"><span class="lab"><font color="Green" title="E38">A80</font></span></a>, <a class="txt" href="#E62:15"><span class="lab"><font color="Green" title="E48">A90</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/>
<a NAME="E65:15"/>
(  not <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/>
<a NAME="E66:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A92</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#T14" onmouseover="rs('measure5/T14')" onmouseout="rh()">MEASURE5:14</a></span>;<br/>
<a NAME="E67:15"/><span class="lab"><font color="Green" title="E51">A93</font></span>: 
<font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E20">A16</font></span></a></span>;<br/>
<a NAME="E68:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A94</font></span>: 
<font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E69:15"/><span class="lab"><font color="Green" title="E53">A95</font></span>: 
<font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:15"><span class="lab"><font color="Green" title="E51">A93</font></span></a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E70:15"/><span class="lab"><font color="Green" title="E54">A96</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E71:15"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">Nseq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E72:15"/><span class="kw">then </span>
<font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">Nseq1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c20">k0</font>,<font color="Maroon" title="c21">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:15"><span class="lab"><font color="Green" title="E39">A81</font></span></a>, <a class="ref" href="valued_0.html#D15" onmouseover="rs('valued_0/D15')" onmouseout="rh()">VALUED_0:def 15</a>, <a class="txt" href="#E46:15"><span class="lab"><font color="Green" title="E37">A79</font></span></a>, <a class="txt" href="#E70:15"><span class="lab"><font color="Green" title="E54">A96</font></span></a></span>;<br/>
<a NAME="E73:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A97</font></span>: 
<font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:15"><span class="lab"><font color="Green" title="E42">A84</font></span></a>, <a class="ref" href="valued_0.html#D15" onmouseover="rs('valued_0/D15')" onmouseout="rh()">VALUED_0:def 15</a></span>;<br/>
<a NAME="E74:15"/>
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#T14" onmouseover="rs('seqm_3/T14')" onmouseout="rh()">SEQM_3:14</a></span>;<br/>
<a NAME="E75:15"/><span class="kw">then </span>
<font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:15"><span class="lab"><font color="Green" title="E42">A84</font></span></a>, <a class="ref" href="valued_0.html#D15" onmouseover="rs('valued_0/D15')" onmouseout="rh()">VALUED_0:def 15</a>, <a class="txt" href="#E70:15"><span class="lab"><font color="Green" title="E54">A96</font></span></a></span>;<br/>
<a NAME="E76:15"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Nseq1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:15"><span class="lab"><font color="Green" title="E44">A86</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E77:15"/>
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:15"><span class="lab"><font color="Green" title="E55">A97</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c24">di</font> =  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>);<br/>
<span class="lab"><font color="Green" title="E56">A98</font></span>: 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default">2</span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p0">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>

;<br/>
<a NAME="E79:15"/>
<font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:15"><span class="lab"><font color="Green" title="E45">A87</font></span></a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E80:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A99</font></span>: 
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:15"><span class="lab"><font color="Green" title="E43">A85</font></span></a>, <a class="txt" href="#E52:15"><span class="lab"><font color="Green" title="E41">A83</font></span></a>, <a class="txt" href="#E68:15"><span class="lab"><font color="Green" title="E52">A94</font></span></a>, <a class="ref" href="jct_misc.html#T8" target="_self" onmouseover="rs('jct_misc/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<a NAME="E81:15"/><span class="lab"><font color="Green" title="E58">A100</font></span>: 
<font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/>
<a NAME="E82:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A101</font></span>: 
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:15"><span class="lab"><font color="Green" title="E40">A82</font></span></a>, <a class="ref" href="jct_misc.html#T7" target="_self" onmouseover="rs('jct_misc/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E60">A102</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_11"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_11_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_11_1"/>
( <font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> or <font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_11_1_1"/>
<font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span><br/><div class="add"><a NAME="E2:15_11_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><a NAME="E3:15_11_1_1"/><span class="kw">then </span>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:15"><span class="lab"><font color="Green" title="E59">A101</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><a NAME="E4:15_11_1_1"/><span class="kw">then </span>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15"><span class="lab"><font color="Green" title="E57">A99</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:15_11_1_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:15"><span class="lab"><font color="Green" title="E49">A91</font></span></a>, <a class="txt" href="#E78:15"><span class="lab"><font color="Green" title="E56">A98</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:15_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A103</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T2" onmouseover="rs('rcomp_1/T2')" onmouseout="rh()">RCOMP_1:2</a></span>;<br/><a NAME="E7:15_11_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:15"><span class="lab"><font color="Green" title="E50">A92</font></span></a>, <a class="txt" href="#E81:15"><span class="lab"><font color="Green" title="E58">A100</font></span></a>, <a class="txt" href="#E49:15"><span class="lab"><font color="Green" title="E40">A82</font></span></a>, <a class="txt" href="#E59:15"><span class="lab"><font color="Green" title="E46">A88</font></span></a>, <a class="txt" href="#E69:15"><span class="lab"><font color="Green" title="E53">A95</font></span></a>, <a class="ref" href="xxreal_2.html#D12" onmouseover="rs('xxreal_2/D12')" onmouseout="rh()">XXREAL_2:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E8:15_11_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_11_1_1"><span class="lab"><font color="Green" title="E61">A103</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_11_1_2"/><span class="lab"><font color="Green" title="E61">A104</font></span>: 
<font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span><br/><div class="add"><a NAME="E2:15_11_1_2"/><span class="lab"><font color="Green" title="E62">A105</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uniform1.html#T11" onmouseover="rs('uniform1/T11')" onmouseout="rh()">UNIFORM1:11</a></span>;<br/><a NAME="E3:15_11_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_11_1_2"><span class="lab"><font color="Green" title="E61">A104</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><a NAME="E4:15_11_1_2"/><span class="kw">then </span>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">SA</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">SB</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:15"><span class="lab"><font color="Green" title="E59">A101</font></span></a>, <a class="txt" href="#E2:15_11_1_2"><span class="lab"><font color="Green" title="E62">A105</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:15_11_1_2"/><span class="kw">then </span>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15"><span class="lab"><font color="Green" title="E57">A99</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:15_11_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:15"><span class="lab"><font color="Green" title="E49">A91</font></span></a>, <a class="txt" href="#E78:15"><span class="lab"><font color="Green" title="E56">A98</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:15_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A106</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T2" onmouseover="rs('rcomp_1/T2')" onmouseout="rh()">RCOMP_1:2</a></span>;<br/><a NAME="E8:15_11_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">sb</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">sa</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:15"><span class="lab"><font color="Green" title="E50">A92</font></span></a>, <a class="txt" href="#E81:15"><span class="lab"><font color="Green" title="E58">A100</font></span></a>, <a class="txt" href="#E49:15"><span class="lab"><font color="Green" title="E40">A82</font></span></a>, <a class="txt" href="#E59:15"><span class="lab"><font color="Green" title="E46">A88</font></span></a>, <a class="txt" href="#E69:15"><span class="lab"><font color="Green" title="E53">A95</font></span></a>, <a class="ref" href="xxreal_2.html#D12" onmouseover="rs('xxreal_2/D12')" onmouseout="rh()">XXREAL_2:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E9:15_11_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15_11_1_2"><span class="lab"><font color="Green" title="E63">A106</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E84:15"/>
<font color="Maroon" title="c6">T</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E85:15"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ga</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">Nseq1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c14">fb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:15"><span class="lab"><font color="Green" title="E60">A102</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
