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<b>let </b><font color="Maroon" title="c1">seq</font>, <font color="Maroon" title="c2">seq1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green" title="E33">A2</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c1">seq</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">seq1</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Olive" title="b1">k</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:45"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">seq1</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:45"><i><font color="Green" title="E33">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">g1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E6:45"/><i><font color="Green" title="E35">A4</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</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="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g1</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:45"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="comseq_2.html#D4" title="COMSEQ_2:def.4">COMSEQ_2:def 4</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">g1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="comseq_2.html#D4" title="COMSEQ_2:def.4">COMSEQ_2:def 4</a><br/>

<b>let </b><font color="Maroon" title="c6">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E7:45"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:45"/><i><font color="Green" title="E36">A5</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c7">n1</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#K17" title="COMPLEX1:func.17">|.</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="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g1</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:45"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c7">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">k</font>;<br/>

<b>let </b><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E10:45"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:45"/><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">l</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">l</font> = <font color="Maroon" title="c10">l</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E14:45"/>
<font color="Maroon" title="c9">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E12:45"><i><font color="Green" title="E38">A7</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">m1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E16:45"/><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon" title="c12">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">k</font>
 ;<br/>
<div><i><font color="Green" title="E40">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1"/>
not <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m1</font>
 ;<br/><a NAME="E2:45_1"/><b>then </b>
<font color="Maroon" title="c12">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:45_1"/>
contradiction
 <b>by </b><i><a class="txt" href="cfcont_1.html#E10:45"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="cfcont_1.html#E16:45"><i><font color="Green" title="E39">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:45"/><i><font color="Green" title="E41">A10</font></i>: 
<font color="Maroon" title="c12">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E16:45"><i><font color="Green" title="E39">A8</font></i></a></i>;<br/>
<a NAME="E19:45"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</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="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g1</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E9:45"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="cfcont_1.html#E17:45"><i><font color="Green" title="E40">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:45"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E4:45"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="cfcont_1.html#E18:45"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="comseq_3.html#D6" title="COMSEQ_3:def.6">COMSEQ_3:def 6</a></i>;<br/>


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