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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="comseq_2.html#V1">bounded</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="comseq_2.html#V1">bounded</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:4"/><i><font color="Green">E6</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="ref" href="comseq_2.html#T8">COMSEQ_2:8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:4"/><i><font color="Green">E7</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="csspace4.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="comseq_2.html#T8">COMSEQ_2:8</a></i>;<br/>
<a NAME="E6:4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <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">c<sub>1</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:4_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">c<sub>1</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</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">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>;<br/>
<a NAME="E2:4_1"/><b>then </b><i><font color="Green">E8</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">c<sub>1</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></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"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></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"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/>
<a NAME="E3:4_1"/>
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="csspace4.html#E3:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="csspace4.html#E5:4"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E4:4_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></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"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:4_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">c<sub>1</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:4_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="comseq_2.html#V1">bounded</a>
 <b>by </b><i><a class="ref" href="comseq_2.html#D3">COMSEQ_2:def 3</a></i>;<br/>


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