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<b>let </b><font color="Maroon">L</font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>, <font color="Maroon">K</font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:90"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E2:90"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <b>and </b><br/><a NAME="E3:90"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:90"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:90_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:90_1"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E3:90_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="ref" href="extreal1.html#T42">EXTREAL1:42</a></i>;<br/><a NAME="E4:90_1"/><b>then </b>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E5:90_1"/>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E6:90"/>
<font color="Maroon">K</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, , <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a>, <a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a></i>;<br/>

<b>thus </b><a NAME="E7:90"/>
( <font color="Maroon">K</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a></i>;<br/>

<a NAME="E8:90"/>
(  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">K</font> &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a>, <a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a></i>;<br/>
<b>hence </b><a NAME="E9:90"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a>, , <a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a></i>;<br/>


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