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<b>let </b><font color="Maroon">X</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">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<br/>



<i><font color="Green">E37</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_2"/><i><font color="Green">E39</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#K1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:36_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>;<br/><b>hence </b><a NAME="E3:36_2"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:36"/>
 <a href="mesfunc2.html#K1">max+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<i><font color="Green">E43</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_4"/><i><font color="Green">E45</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">max-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:36_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/><a NAME="E3:36_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="mesfunc2.html#T10">MESFUNC2:10</a></i>;<br/><a NAME="E4:36_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K3">max-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E5:36_4"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:36"/>
 <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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