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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="supinf_1.html#K3">REAL</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon">f</font> is <a href="mesfunc5.html#V2">nonpositive</a>
 ;<br/><a NAME="E2:60_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="mesfunc5.html#V1">nonpositive</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2">MESFUNC5:def 2</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:60_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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/><a NAME="E3:60_1_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><b>hence </b><a NAME="E4:60_1_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="mesfunc5.html#D1">MESFUNC5:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:60_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <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>

<b>assume </b><a NAME="E2:60"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<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">MESFUNC5:def 1</a>,<a class="ref" href="mesfunc5.html#D2">MESFUNC5:def 2</a><br/>

<b>assume </b><a NAME="E3:60"/>
<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="E4:60"/><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="seq_1.html#K2">.</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="E5:60"/>
not 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


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