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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:92_1"/>
<font color="Maroon" title="c2">F</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 ;<br/><a NAME="E2:92_1"/><b>then </b><i><font color="Green" title="E67">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">F</font> is <a href="supinf_2.html#V5" title="SUPINF_2:attr.5">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#D22" target="_self" title="SUPINF_2:def.22">Def22</a></i>;<br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_1"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> or  not <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:92_1_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:92_1"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="supinf_2.html#D15" target="_self" title="SUPINF_2:def.15">Def15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_2"><b>suppose </b></a><a NAME="E1:92_1_1_2"/>
not <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:92_1_1_2"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">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="E2:92"/><i><font color="Green" title="E67">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b1">n</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">y</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="supinf_2.html#D15" target="_self" title="SUPINF_2:def.15">SUPINF_2:def 15</a>,<a class="ref" href="supinf_2.html#D22" target="_self" title="SUPINF_2:def.22">SUPINF_2:def 22</a><br/>

<b>assume </b><a NAME="E3:92"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:92"/><i><font color="Green" title="E68">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:92"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:92"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="supinf_2.html#E5:92"><i><font color="Green" title="E68">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:92"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="supinf_2.html#E5:92"><i><font color="Green" title="E68">A3</font></i></a></i>;<br/>


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