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<b>let </b><font color="Maroon" title="c1">X</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">X</font>, <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/>



<i><font color="Green" title="E16">A1</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></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="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:49_2"/><i><font color="Green" title="E17">A2</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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:49_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>;<br/><a NAME="E3:49_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 <b>by </b><i><a class="ref" href="mesfunc6.html#T43" target="_self" title="MESFUNC6:th.43">Th43</a></i>;<br/><b>hence </b><a NAME="E4:49_2"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="mesfunc6.html#E1:49"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="mesfunc6.html#E1:49_2"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:49"/>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a>, <a class="txt" href="mesfunc6.html#E1:49"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>


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