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<b>let </b><font color="Maroon">L</font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:84"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>and </b><br/><a NAME="E2:84"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <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">K</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:84_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:84_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:84_1"/>
<font color="Maroon">z</font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:84_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="mesfunc5.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:84"/><b>then </b>
<font color="Maroon">K</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<a NAME="E5:84"/><b>then </b><i><font color="Green">E40</font></i>: 
 <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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>
<a NAME="E6:84"/>
<font color="Maroon">K</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>
<b>hence </b><a NAME="E7:84"/>
 <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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>


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