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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">F</font> is <a href="mesfunc5.html#V2" target="_self">nonpositive</a>
 ;<br/><a NAME="E2:16_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> is <a href="mesfunc5.html#V1" target="_self">nonpositive</a>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:16_1_1_1"/>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_2"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_2"/>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:16"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>  holds  <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/>

<b>let </b><font color="Maroon">y</font> be   <a href="supinf_1.html#NM3">R_eal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="mesfunc5.html#D1" target="_self">MESFUNC5:def 1</a>,<a class="ref" href="mesfunc5.html#D2" target="_self">MESFUNC5:def 2</a><br/>

<b>assume </b><a NAME="E3:16"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E4:16"/><b>then </b>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:16"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i/>;<br/>


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