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<b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">seq1</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> &amp; <font color="Maroon">seq2</font> is <a href="seq_2.html#V2">bounded_below</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">M</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="E3:34"/><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">M</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc1.html#D4" target="_self">LIMFUNC1:def 4</a><br/>

<a NAME="E4:34"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">M</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:34"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E7:34"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E8:34"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:34"/>
<font color="Maroon">M</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:34"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">M</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E11:34"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


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