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<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:86"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>and </b><br/><a NAME="E2:86"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</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="E4:86"/><i><font color="Green">E34</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>and </b><br/><a NAME="E5:86"/><i><font color="Green">E37</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">r1</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>
<b>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="E7:86"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font>
 <b>and </b><br/><a NAME="E8:86"/><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  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</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">r2</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D7" target="_self">Def7</a>, <a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r2</font>;<br/><a NAME="E1:86_1"/>
0 <a href="nat_1.html#K2">*</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T7" target="_self">Th7</a>, <a class="ref" href="seqm_3.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T98">XREAL_1:98</a></i>;<br/><b>hence </b><a NAME="E2:86_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</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="E3:86_1"/><i><font color="Green">E42</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">r1</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E4:86_1"/><i><font color="Green">E64</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</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">r2</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E5:86_1"/><i><font color="Green">E65</font></i>: 
0 <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="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E6:86_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E7:86_1"/><b>then </b>
<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="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T12" target="_self">Th12</a>, <a class="ref" href="seqm_3.html#T13" target="_self">Th13</a>, <a class="ref" href="seqm_3.html#T14" target="_self">Th14</a>, <a class="ref" href="xreal_1.html#T98">XREAL_1:98</a></i>;<br/><a NAME="E8:86_1"/><b>then </b>
 <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="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/><b>hence </b><a NAME="E9:86_1"/>
 <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#K11">(#)</a> <font color="Maroon">seq1</font></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> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:86"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>

<a NAME="E11:86"/>
 <a href="seq_1.html#K17">-</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D7" target="_self">Def7</a>, <a class="txt" href="seqm_3.html#E13"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<a NAME="E12:86"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">seq1</font></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i>, <a class="ref" href="seqm_3.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E13:86"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>


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