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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>thus </b><a NAME="E1:7"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V3" target="_self">bounded</a> implies  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/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <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="Olive">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3" target="_self">bounded</a>
 ;<br/>

<a NAME="E2:7_1"/><b>then </b>
<font color="Maroon">seq</font> is <a href="seq_2.html#V1" target="_self">bounded_above</a>
 
<b>by </b><i/>;<br/>
<b>then 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="E4:7_1"/><i><font color="Green">E38</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">seq</font> <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">r1</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:7_1"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V2" target="_self">bounded_below</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">r2</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="E7:7_1"/><i><font color="Green">E39</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">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r1</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">r2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1;<br/>

<a NAME="E8:7_1"/><i><font color="Green">E40</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E9:7_1"/><i><font color="Green">E41</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E10:7_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</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">r1</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">r2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E11:7_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>

<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/>

<a NAME="E12:7_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E13:7_1"/>
<font color="Maroon">seq</font> <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">r1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:7_1"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:7_1"/><b>then </b>
<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="real_1.html#K3">+</a> 0 <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">r1</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">r2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E16:7_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">seq</font> <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">g</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E17:7_1"/><i><font color="Green">E44</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E18:7_1"/><i><font color="Green">E45</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E19:7_1"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E20:7_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E21:7_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</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="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E22:7_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <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#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E23:7_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">g</font>
 
;<br/>
<b>hence </b><a NAME="E24:7_1"/>
 <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#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">r</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="E2:7"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green">E53</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  <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="Olive">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">g</font> = <font color="Maroon">r</font>;<br/><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/><a NAME="E1:7_2"/><i><font color="Green">E54</font></i>: 
 <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#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><a NAME="E2:7_2"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><b>hence </b><a NAME="E3:7_2"/>
<font color="Maroon">seq</font> <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">g</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="E5:7"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V1" target="_self">bounded_above</a>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D5" target="_self">SEQ_2:def 5</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">g</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font>;<br/><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/><a NAME="E1:7_3"/>
 <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#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E2:7_3"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</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="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:7_3"/>
 <a href="real_1.html#K1">-</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> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><b>hence </b><a NAME="E4:7_3"/>
<font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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="E7:7"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V2" target="_self">bounded_below</a>
 <b>by </b><i/>;<br/>


</div>
