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<div class="add">

<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green">E32</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>given </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><a NAME="E4:13"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a>,<a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a><br/>

<b>given </b><font color="Maroon">b</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><a NAME="E5:13"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a><br/>

<b>set </b><font color="Maroon">aa</font> =  <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">ab</font> =  <a href="complex1.html#K18">abs</a> <font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">rab</font> = 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span>)</span>;<br/>
<a NAME="E6:13"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:13"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E8:13"/><b>then </b><i><font color="Green">E40</font></i>: 
0 <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1"/>
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span>
 ;<br/><a NAME="E2:13_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:13_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:13_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E10:13"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E11:13"/><b>then </b><i><font color="Green">E41</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">s</font> = <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">At</font> =  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>;<br/><a NAME="E1:13_2"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E2:13_2"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:13_2"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K2">"</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>;<br/><a NAME="E4:13_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a>, <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/><a NAME="E5:13_2"/><b>then </b><i><font color="Green">E46</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><a NAME="E6:13_2"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/><a NAME="E7:13_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T15">ABSVALUE:15</a></i>;<br/><a NAME="E8:13_2"/>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/><b>hence </b><a NAME="E9:13_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:13"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E15:13"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:13"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
