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<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="E1:43"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V4">non-increasing</a>
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a>
 ;<br/>

<b>consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:43"/><i><font color="Green">E48</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">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1">Real</a>] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:43"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] )
 <b>from </b><i><a class="ref" href="real_1.html#S1">REAL_1:sch 1</a>();<br/></i>
<a NAME="E7:43"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:43"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">X</font> is <a href="seq_4.html#V2" target="_self">bounded_below</a> iff  ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> )
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r1</font>;<br/><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:43_1"/><b>then </b>
 ex <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:43_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>take </b><font color="Maroon">g</font> =  <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a><br/>

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E10:43"/><i><font color="Green">E53</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E12:43"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E13:43"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">n1</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="E15:43"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T8" target="_self">Th8</a></i>;<br/>
<b>take </b><font color="Maroon">n</font> = <font color="Maroon">n1</font>;<br/>

<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> ;<br/>

<b>assume </b><a NAME="E16:43"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E17:43"/><b>then </b>
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="seqm_3.html#T14">SEQM_3:14</a></i>;<br/>
<a NAME="E18:43"/><b>then </b>
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E1:43"><i><font color="Green">E45</font></i></a>, <a class="txt" href="seq_4.html#E2:43"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:43"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E20:43"/>
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E21:43"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="seq_4.html#E3"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/>
<a NAME="E22:43"/><b>then </b><i><font color="Green">E58</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E23:43"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="txt" href="seq_4.html#E4"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E24:43"/>
 <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="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T9" target="_self">Th9</a>, <a class="txt" href="seq_4.html#E4:43"><i><font color="Green">E48</font></i></a>, <a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>


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