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<b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <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">C</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>





<a NAME="E1:18"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_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> or  not <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>suppose </b></a><a NAME="E1:18_1_1"/><i><font color="Green">E34</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>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:18_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E3:18_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:18_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="extreal2.html#K1">max</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E5:18_1_1"/><b>then </b>
(  not  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a>  or  not  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, <a class="ref" href="extreal2.html#T96">EXTREAL2:96</a></i>;<br/><a NAME="E6:18_1_1"/><b>then </b>
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3">-</a> <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/><a NAME="E7:18_1_1"/><b>then </b>
 <a href="extreal2.html#K1">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>,<a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T24">EXTREAL1:24</a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E8:18_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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="18_1_2"><b>suppose </b></a><a NAME="E1:18_1_2"/>
not <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/><div class="add"><b>hence </b><a NAME="E2:18_1_2"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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>

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