<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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="c1">seq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  not <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E2:43"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a> 
 <span class="kw">and </span><br/><a NAME="E3:43"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> </span><br/>

<span class="kw">given </span><font color="Maroon" title="c2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><a NAME="E5:43"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="seq_2.html#D3" onmouseover="rs('seq_2/D3')" onmouseout="rh()">SEQ_2:def 3</a>,<a class="ref" href="seq_2.html#D8" onmouseover="rs('seq_2/D8')" onmouseout="rh()">SEQ_2:def 8</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="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> </span><br/>

<span class="kw">given </span><font color="Maroon" title="c3">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><a NAME="E7:43"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="seq_2.html#D4" onmouseover="rs('seq_2/D4')" onmouseout="rh()">SEQ_2:def 4</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"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">aa</font> = <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">ab</font> = <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">rab</font> = 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></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="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span>;<br/>
<a NAME="E8:43"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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"><font color="Maroon" title="c2">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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"><font color="Maroon" title="c3">s</font></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#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E24">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><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">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <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="c1">seq</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></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="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">n</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"> <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="c1">seq</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></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="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span></span><br/><span class="kw">set </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c9">t</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c10">At</font> = <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="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>;<br/><a NAME="E1:43_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T10" onmouseover="rs('valued_1/T10')" onmouseout="rh()">VALUED_1:10</a></span>;<br/><a NAME="E2:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A8</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="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</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="absvalue.html#T7" onmouseover="rs('absvalue/T7')" onmouseout="rh()">ABSVALUE:7</a></span>;<br/><a NAME="E3:43_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T10" onmouseover="rs('valued_1/T10')" onmouseout="rh()">VALUED_1:10</a></span>;<br/><a NAME="E4:43_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E2:43"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="seq_1.html#T5" onmouseover="rs('seq_1/T5')" onmouseout="rh()">SEQ_1:5</a>, <a class="ref" href="xcmplx_1.html#T202" onmouseover="rs('xcmplx_1/T202')" onmouseout="rh()">XCMPLX_1:202</a></span>;<br/><a NAME="E5:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A9</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</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="complex1.html#T47" onmouseover="rs('complex1/T47')" onmouseout="rh()">COMPLEX1:47</a></span>;<br/><a NAME="E6:43_1"/>
( <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/><a NAME="E7:43_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"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font></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"><font color="Maroon" title="c2">r</font></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"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T37" target="_self" onmouseover="rs('measure6/T37')" onmouseout="rh()">Th37</a></span>;<br/><a NAME="E8:43_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"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></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="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="xreal_1.html#T88" onmouseover="rs('xreal_1/T88')" onmouseout="rh()">XREAL_1:88</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_1"/>
<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="c1">seq</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></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="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E25">A8</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>
<a NAME="E10:43"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E21">A4</font></span></a></span>;<br/>
<a NAME="E11:43"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2"><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  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></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"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_2"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></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"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</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"> contradiction</span><br/><a NAME="E2:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></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"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/><a NAME="E3:43_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T36" target="_self" onmouseover="rs('measure6/T36')" onmouseout="rh()">Th36</a></span>;<br/><span class="kw">hence </span><a NAME="E4:43_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="measure6.html#T36" target="_self" onmouseover="rs('measure6/T36')" onmouseout="rh()">Th36</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="c7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E14:43"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</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"><font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></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="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E20">A3</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="E15:43"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c7">n</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="E16:43"/>
<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="c1">seq</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">r</font></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="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43"><span class="lab"><font color="Green" title="E27">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:43"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E28">A14</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>
