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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="E1:105_1"/>
<font color="Maroon">F</font> is <a href="supinf_2.html#V6" target="_self">nonnegative</a>
 ;<br/><a NAME="E2:105_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> is <a href="supinf_2.html#V5" target="_self">nonnegative</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="105_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_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="105_1_1_1"><b>suppose </b></a><a NAME="E1:105_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="E2:105_1_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</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="E3:105_1_1_1"/>
 <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#D5" target="_self">Def5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_2"><b>suppose </b></a><a NAME="E1:105_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="E2:105_1_1_2"/>
 <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 <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="E2:105"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>  holds   <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">n</font>
 ;<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="supinf_2.html#D15" target="_self">SUPINF_2:def 15</a>,<a class="ref" href="supinf_2.html#D22" target="_self">SUPINF_2:def 22</a><br/>

<b>assume </b><a NAME="E3:105"/>
<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/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:105"/><i><font color="Green">E29</font></i>: 
( <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> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:105"/>
 <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E7:105"/>
 <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


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