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<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">C</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/>



<i><font color="Green" title="E14">A1</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="ref" href="mesfunc2.html#D3" target="_self" title="MESFUNC2:def.3">Def3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></i>
;<br/>
<a NAME="E2:17"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2" target="_self" title="MESFUNC2:def.2">Def2</a></i>;<br/>
<a NAME="E3:17"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/>

<a NAME="E2:17_2"/><b>then </b><i><font color="Green" title="E17">A4</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>,<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D3" target="_self" title="MESFUNC2:def.3">Def3</a></i>;<br/>
<a NAME="E3:17_2"/>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc2.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="mesfunc2.html#E1:17_2"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:17_2"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="mesfunc2.html#E2:17"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="mesfunc2.html#E1:17_2"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="mesfunc2.html#E2:17_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="mesfunc2.html#D2" target="_self" title="MESFUNC2:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:17"/>
 <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc2.html#E2:17"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


</div>
