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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</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">s1</font>, <font color="Olive" title="b2">s2</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">s1</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp; <font color="Olive" title="b2">s2</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Olive" title="b1">s1</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>  &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b1">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</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>  &amp; <font color="Olive" title="b2">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font> ) ) holds <br/> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Olive" title="b2">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Olive" title="b1">s1</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">s1</font>, <font color="Maroon" title="c3">s2</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="c2">s1</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp; <font color="Maroon" title="c3">s2</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</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>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</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>  &amp; <font color="Maroon" title="c3">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font> ) ) implies  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:110"/><span class="lab"><font color="Green" title="E93">A1</font></span>: 
<font color="Maroon" title="c2">s1</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E2:110"/><span class="lab"><font color="Green" title="E94">A2</font></span>: 
<font color="Maroon" title="c3">s2</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E3:110"/><span class="lab"><font color="Green" title="E95">A3</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</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">and </span><br/><a NAME="E4:110"/><span class="lab"><font color="Green" title="E96">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</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>  &amp; <font color="Maroon" title="c3">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</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">  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_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="E1:110_1"/>
( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1"><span class="kw">suppose </span></a><a NAME="E1:110_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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">  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:110_1_1"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E93">A1</font></span></a>, <a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E95">A3</font></span></a>, <a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E96">A4</font></span></a>, <a class="txt" href="prepower.html#E105"><span class="lab"><font color="Green" title="E91">Lm10</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="110_1_2"><span class="kw">suppose </span></a><a NAME="E1:110_1_2"/><span class="lab"><font color="Green" title="E97">A5</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span><br/><div class="add"><a NAME="E2:110_1_2"/>
<font color="Maroon" title="c2">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E93">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">d</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:110_1_2"/><span class="lab"><font color="Green" title="E98">A6</font></span>: 
<font color="Maroon" title="c4">d</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">and </span><br/><a NAME="E5:110_1_2"/><span class="lab"><font color="Green" title="E99">A7</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"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T3" onmouseover="rs('seq_2/T3')" onmouseout="rh()">SEQ_2:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">d</font> = <font color="Maroon" title="c4">d</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><a NAME="E7:110_1_2"/><span class="lab"><font color="Green" title="E100">A8</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</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="#E4:110_1_2"><span class="lab"><font color="Green" title="E98">A6</font></span></a>, <a class="ref" href="prepower.html#T52" target="_self" onmouseover="rs('prepower/T52')" onmouseout="rh()">Th52</a></span>;<br/><div><span class="lab"><font color="Green" title="E101">A9</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_2_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">  not  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:110_1_2_1"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:110_1_2_1"/><span class="lab"><font color="Green" title="E102">A10</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="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</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="c3">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> <font color="Maroon" title="c5">d</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="txt" href="#E7:110_1_2"><span class="lab"><font color="Green" title="E100">A8</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/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_2_1_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">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds  not <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">m</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">  not <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</font></span><br/><a NAME="E1:110_1_2_1_1"/><span class="lab"><font color="Green" title="E103">A11</font></span>: 
<font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</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="#E4:110"><span class="lab"><font color="Green" title="E96">A4</font></span></a></span>;<br/><a NAME="E2:110_1_2_1_1"/><span class="lab"><font color="Green" title="E104">A12</font></span>: 
<font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</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="#E4:110"><span class="lab"><font color="Green" title="E96">A4</font></span></a></span>;<br/><a NAME="E3:110_1_2_1_1"/><span class="lab"><font color="Green" title="E105">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</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="#E4:110"><span class="lab"><font color="Green" title="E96">A4</font></span></a>, <a class="ref" href="prepower.html#T52" target="_self" onmouseover="rs('prepower/T52')" onmouseout="rh()">Th52</a></span>;<br/><a NAME="E4:110_1_2_1_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="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</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> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_2"><span class="lab"><font color="Green" title="E99">A7</font></span></a></span>;<br/><a NAME="E5:110_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_2_1_1"><span class="lab"><font color="Green" title="E103">A11</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><a NAME="E6:110_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">d</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_2_1_1"><span class="lab"><font color="Green" title="E103">A11</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E7:110_1_2_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">d</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_2_1_1"><span class="lab"><font color="Green" title="E104">A12</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>;<br/><a NAME="E8:110_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_2"><span class="lab"><font color="Green" title="E97">A5</font></span></a>, <a class="ref" href="prepower.html#T61" target="_self" onmouseover="rs('prepower/T61')" onmouseout="rh()">Th61</a></span>;<br/><a NAME="E9:110_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_2"><span class="lab"><font color="Green" title="E98">A6</font></span></a>, <a class="txt" href="#E1:110_1_2_1_1"><span class="lab"><font color="Green" title="E103">A11</font></span></a>, <a class="ref" href="prepower.html#T58" target="_self" onmouseover="rs('prepower/T58')" onmouseout="rh()">Th58</a></span>;<br/><a NAME="E10:110_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A14</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></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"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_2_1_1"><span class="lab"><font color="Green" title="E105">A13</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E11:110_1_2_1_1"/><span class="lab"><font color="Green" title="E107">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</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="#E4:110"><span class="lab"><font color="Green" title="E96">A4</font></span></a>, <a class="ref" href="prepower.html#T52" target="_self" onmouseover="rs('prepower/T52')" onmouseout="rh()">Th52</a></span>;<br/><span class="kw">assume </span><a NAME="E12:110_1_2_1_1"/>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</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/><a NAME="E13:110_1_2_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"><font color="Maroon" title="c3">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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> <font color="Maroon" title="c5">d</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_2_1"><span class="lab"><font color="Green" title="E102">A10</font></span></a></span>;<br/><a NAME="E14:110_1_2_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"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</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> <font color="Maroon" title="c5">d</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E96">A4</font></span></a></span>;<br/><a NAME="E15:110_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">d</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_2_1_1"><span class="lab"><font color="Green" title="E105">A13</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E16:110_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:110_1_2_1_1"><span class="lab"><font color="Green" title="E107">A15</font></span></a>, <a class="txt" href="#E10:110_1_2_1_1"><span class="lab"><font color="Green" title="E106">A14</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</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">hence </span><a NAME="E5:110_1_2_1"/>
contradiction
 ; <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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_2_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">  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">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">n</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"> <font color="Maroon" title="c3">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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><br/><a NAME="E1:110_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</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="#E4:110"><span class="lab"><font color="Green" title="E96">A4</font></span></a>, <a class="ref" href="prepower.html#T52" target="_self" onmouseover="rs('prepower/T52')" onmouseout="rh()">Th52</a></span>;<br/><span class="kw">hence </span><a NAME="E2:110_1_2_2"/>
<font color="Maroon" title="c3">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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="#E4:110"><span class="lab"><font color="Green" title="E96">A4</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="E10:110_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A16</font></span>: 
<font color="Maroon" title="c3">s2</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T5" onmouseover="rs('seq_1/T5')" onmouseout="rh()">SEQ_1:5</a></span>;<br/><a NAME="E11:110_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A17</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</font></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="#E2:110"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="txt" href="#E8:110_1_2"><span class="lab"><font color="Green" title="E101">A9</font></span></a>, <a class="ref" href="seq_2.html#T22" onmouseover="rs('seq_2/T22')" onmouseout="rh()">SEQ_2:22</a></span>;<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="hidden.html#M1" title="HIDDEN:mode.1">object</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> ;<br/><span class="kw">consider </span><font color="Maroon" title="c6">s</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="E13:110_1_2"/><span class="lab"><font color="Green" title="E104">A18</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="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="seq_1.html#S1" onmouseover="rs('seq_1/S1')" onmouseout="rh()">SEQ_1:sch 1</a>();<br/></span><div><span class="lab"><font color="Green" title="E105">A19</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_2_3"><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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</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>  &amp; <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">p</font></span>)</span> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">n</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"> ( <font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</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>  &amp; <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">p</font></span>)</span> )</span><br/><font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></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="#E13:110_1_2"><span class="lab"><font color="Green" title="E104">A18</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></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="#E4:110"><span class="lab"><font color="Green" title="E96">A4</font></span></a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E96">A4</font></span></a>, <a class="ref" href="prepower.html#T54" target="_self" onmouseover="rs('prepower/T54')" onmouseout="rh()">Th54</a></span>
;<br/><span class="kw">hence </span><a NAME="E2:110_1_2_3"/>
( <font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</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>  &amp; <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">p</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E96">A4</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="E15:110_1_2"/><span class="lab"><font color="Green" title="E106">A20</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">s2</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="E16:110_1_2"/><span class="lab"><font color="Green" title="E107">A21</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">s</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="E17:110_1_2"/>
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">s</font> holds <br/><font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></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="#E13:110_1_2"><span class="lab"><font color="Green" title="E104">A18</font></span></a></span>;<br/><a NAME="E18:110_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A22</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:110_1_2"><span class="lab"><font color="Green" title="E107">A21</font></span></a>, <a class="txt" href="#E15:110_1_2"><span class="lab"><font color="Green" title="E106">A20</font></span></a>, <a class="ref" href="valued_1.html#D7" onmouseover="rs('valued_1/D7')" onmouseout="rh()">VALUED_1:def 7</a></span>;<br/><a NAME="E19:110_1_2"/>
<font color="Maroon" title="c3">s2</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  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="#E2:110"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="txt" href="#E10:110_1_2"><span class="lab"><font color="Green" title="E102">A16</font></span></a>, <a class="txt" href="#E8:110_1_2"><span class="lab"><font color="Green" title="E101">A9</font></span></a>, <a class="ref" href="seq_2.html#T21" onmouseover="rs('seq_2/T21')" onmouseout="rh()">SEQ_2:21</a></span>;<br/><a NAME="E20:110_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</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"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E93">A1</font></span></a>, <a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E95">A3</font></span></a>, <a class="txt" href="#E1:110_1_2"><span class="lab"><font color="Green" title="E97">A5</font></span></a>, <a class="txt" href="#E11:110_1_2"><span class="lab"><font color="Green" title="E103">A17</font></span></a>, <a class="txt" href="#E18:110_1_2"><span class="lab"><font color="Green" title="E108">A22</font></span></a>, <a class="txt" href="#E14:110_1_2"><span class="lab"><font color="Green" title="E105">A19</font></span></a>, <a class="txt" href="prepower.html#E105"><span class="lab"><font color="Green" title="E91">Lm10</font></span></a></span>;<br/><a NAME="E21:110_1_2"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</font></span>)</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">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E95">A3</font></span></a>, <a class="ref" href="prepower.html#T54" target="_self" onmouseover="rs('prepower/T54')" onmouseout="rh()">Th54</a></span>;<br/><span class="kw">hence </span><a NAME="E22:110_1_2"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T59" onmouseover="rs('xcmplx_1/T59')" onmouseout="rh()">XCMPLX_1:59</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>
