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<b>let </b><font color="Maroon" title="c1">seq</font>, <font color="Maroon" title="c2">seq1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>thus </b><a NAME="E1:85"/><i><font color="Green" title="E53">A1</font></i>: 
( <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> implies <font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">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" title="E54">A2</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 <b>and </b><br/><a NAME="E2:85_1"/><i><font color="Green" title="E55">A3</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">r1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:85_1"/><i><font color="Green" title="E56">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r1</font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E1:85_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="seq_2.html#D3" title="SEQ_2:def.3">SEQ_2:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">r2</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E6:85_1"/><i><font color="Green" title="E57">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E2:85_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="seq_2.html#D3" title="SEQ_2:def.3">SEQ_2:def 3</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c3">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D3" title="SEQ_2:def.3">SEQ_2:def 3</a><br/>

<b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E7:85_1"/><i><font color="Green" title="E58">A6</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r1</font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E4:85_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/>
<a NAME="E8:85_1"/>
<font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r2</font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E6:85_1"><i><font color="Green" title="E57">A5</font></i></a></i>;<br/>
<a NAME="E9:85_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r2</font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E7:85_1"><i><font color="Green" title="E58">A6</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:85_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/>


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

<b>thus </b><a NAME="E2:85"/><i><font color="Green" title="E54">A7</font></i>: 
( <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> implies <font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">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" title="E55">A8</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>
 <b>and </b><br/><a NAME="E2:85_2"/><i><font color="Green" title="E56">A9</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">r1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:85_2"/><i><font color="Green" title="E57">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E1:85_2"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="seq_2.html#D4" title="SEQ_2:def.4">SEQ_2:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">r2</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E6:85_2"/><i><font color="Green" title="E58">A11</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E2:85_2"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="seq_2.html#D4" title="SEQ_2:def.4">SEQ_2:def 4</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c3">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D4" title="SEQ_2:def.4">SEQ_2:def 4</a><br/>

<b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E7:85_2"/><i><font color="Green" title="E59">A12</font></i>: 
<font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E4:85_2"><i><font color="Green" title="E57">A10</font></i></a></i>;<br/>
<a NAME="E8:85_2"/>
<font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E6:85_2"><i><font color="Green" title="E58">A11</font></i></a></i>;<br/>
<a NAME="E9:85_2"/><b>then </b>
<font color="Maroon" title="c3">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E7:85_2"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:85_2"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">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" title="E55">A13</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 <b>and </b><br/><a NAME="E4:85"/><i><font color="Green" title="E56">A14</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 ;<br/>

<b>thus </b><a NAME="E5:85"/>
<font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 <b>by </b><i><a class="txt" href="seqm_3.html#E1:85"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="seqm_3.html#E3:85"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="seqm_3.html#E4:85"><i><font color="Green" title="E56">A14</font></i></a>, <a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a><br/>

<b>thus </b><a NAME="E6:85"/>
<font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c2">seq1</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>
 <b>by </b><i><a class="txt" href="seqm_3.html#E2:85"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="seqm_3.html#E3:85"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="seqm_3.html#E4:85"><i><font color="Green" title="E56">A14</font></i></a>, <a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/>


</div>
