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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>thus </b><a NAME="E1:85"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V1">bounded_above</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V1">bounded_above</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:85_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>and </b><br/><a NAME="E2:85_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></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="E4:85_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E1:85_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></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="E6:85_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E2:85_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E7:85_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E4:85_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E8:85_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E6:85_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E9:85_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E7:85_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:85_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


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

<b>thus </b><a NAME="E2:85"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:85_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V2">bounded_below</a>
 <b>and </b><br/><a NAME="E2:85_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></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="E4:85_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E1:85_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></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="E6:85_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E2:85_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E7:85_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E4:85_2"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E8:85_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E6:85_2"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E9:85_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E7:85_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:85_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E3:85"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a>
 <b>and </b><br/><a NAME="E4:85"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<b>thus </b><a NAME="E5:85"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i><a class="txt" href="seqm_3.html#E1:85"><i><font color="Green">E53</font></i></a>, <a class="txt" href="seqm_3.html#E3:85"><i><font color="Green">E55</font></i></a>, <a class="txt" href="seqm_3.html#E4:85"><i><font color="Green">E56</font></i></a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a><br/>

<b>thus </b><a NAME="E6:85"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a>
 <b>by </b><i><a class="txt" href="seqm_3.html#E2:85"><i><font color="Green">E54</font></i></a>, <a class="txt" href="seqm_3.html#E3:85"><i><font color="Green">E55</font></i></a>, <a class="txt" href="seqm_3.html#E4:85"><i><font color="Green">E56</font></i></a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>


</div>
