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<b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:60"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">seq</font> is <a href="comseq_2.html#V2">convergent</a>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E2:60"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:60"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/>
<b>then 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="E5:60"/><i><font color="Green">E37</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/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:60_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> )
 ;<br/><a NAME="E2:60_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="comseq_3.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E3:60_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="comseq_3.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E4:60_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="comseq_3.html#E6"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:60_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T143">COMPLEX1:143</a></i>;<br/><b>hence </b><a NAME="E6:60_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:60"/>
for <font color="Olive">m</font>, <font color="Olive">l</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> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">l</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>


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