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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b4">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font>  st <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">E</font> &amp; <font color="Olive" title="b2">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b3">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font> &amp; <font color="Olive" title="b3">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font> holds <br/><font color="Olive" title="b2">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Olive" title="b3">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font></span><br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b3">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">E</font> &amp; <font color="Olive" title="b1">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b2">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b1">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font> &amp; <font color="Olive" title="b2">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font> holds <br/><font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Olive" title="b2">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</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#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp; <font color="Maroon" title="c3">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font></span><br/><b>let </b><font color="Maroon" title="c5">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> &amp; <font color="Maroon" title="c3">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> implies <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )</span><br/>









<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E19">A1</font></i>: 
( <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> &amp; <font color="Maroon" title="c3">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font></span><br/>

<a NAME="E2:31"/><i><font color="Green" title="E20">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</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"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</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"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> )
 
<b>by </b><i><a class="ref" href="mesfunc7.html#D5" target="_self" title="MESFUNC7:def.5">Def5</a></i>;<br/>
<a NAME="E3:31"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</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"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</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"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green" title="E22">A4</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="mesfunc2.html#T2" title="MESFUNC2:th.2">MESFUNC2:2</a></i>;<br/>
<a NAME="E5:31"/><b>then </b><i><font color="Green" title="E23">A5</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green" title="E24">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a></i>;<br/>
<a NAME="E7:31"/><i><font color="Green" title="E25">A7</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></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"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></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"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E8:31"/><i><font color="Green" title="E26">A8</font></i>: 
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">X</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"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )</span><br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c6">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"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><br/>

<a NAME="E2:31_1"/><b>then </b>
( <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E5:31"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">c1</font> = <font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c8">c2</font> = <font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="extreal2.html#T102" title="EXTREAL2:th.102">EXTREAL2:102</a></i>;<br/>
<a NAME="E4:31_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="mesfunc7.html#K5" title="MESFUNC7:func.5">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:31_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="mesfunc7.html#D5" target="_self" title="MESFUNC7:def.5">Def5</a></i>;<br/>
<a NAME="E5:31_1"/><b>then </b><i><font color="Green" title="E28">A10</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="mesfunc7.html#K5" title="MESFUNC7:func.5">|^</a> 1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#T10" target="_self" title="MESFUNC7:th.10">Th10</a></i>;<br/>
<a NAME="E6:31_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/>
<a NAME="E7:31_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T49" title="EXTREAL2:th.49">EXTREAL2:49</a></i>;<br/>
<a NAME="E8:31_1"/><b>then </b><i><font color="Green" title="E29">A11</font></i>: 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/>
<a NAME="E9:31_1"/>
( <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> &amp; <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> )
 
<b>by </b><i><a class="txt" href="#E2:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="#E1:31_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<b>then </b><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> = 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p3"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<b>by </b><i><a class="txt" href="#E5:31_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="mesfunc7.html#T9" target="_self" title="MESFUNC7:th.9">Th9</a></i>
<br/>.= 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="extreal2.html#T56" title="EXTREAL2:th.56">EXTREAL2:56</a></i>
<br/>.= 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="extreal2.html#T70" title="EXTREAL2:th.70">EXTREAL2:70</a></i>
<br/>.= 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="extreal2.html#T56" title="EXTREAL2:th.56">EXTREAL2:56</a></i>
;<br/>
<b>hence </b><a NAME="E11:31_1"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="#E8:31_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:31"/>
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">X</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"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )</span><br/>

<b>assume </b><a NAME="E1:31_2"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c6">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"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><br/>

<a NAME="E2:31_2"/><b>then </b>
( <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E5:31"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">c1</font> = <font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c8">c2</font> = <font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="extreal2.html#T102" title="EXTREAL2:th.102">EXTREAL2:102</a></i>;<br/>
<a NAME="E4:31_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="mesfunc7.html#K5" title="MESFUNC7:func.5">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:31_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="mesfunc7.html#D5" target="_self" title="MESFUNC7:def.5">Def5</a></i>;<br/>
<a NAME="E5:31_2"/><b>then </b><i><font color="Green" title="E28">A13</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="mesfunc7.html#K5" title="MESFUNC7:func.5">|^</a> 1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#T10" target="_self" title="MESFUNC7:th.10">Th10</a></i>;<br/>
<a NAME="E6:31_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<a NAME="E7:31_2"/><b>then </b>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T49" title="EXTREAL2:th.49">EXTREAL2:49</a></i>;<br/>
<a NAME="E8:31_2"/><b>then </b><i><font color="Green" title="E29">A14</font></i>: 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/>
<a NAME="E9:31_2"/>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 
<b>by </b><i><a class="txt" href="#E2:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="#E1:31_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E10:31_2"/><b>then </b>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 
<b>by </b><i><a class="txt" href="#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="#E1:31_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<b>then </b><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> = 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p3"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<b>by </b><i><a class="txt" href="#E5:31_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="mesfunc7.html#T9" target="_self" title="MESFUNC7:th.9">Th9</a></i>
<br/>.= 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="extreal2.html#T56" title="EXTREAL2:th.56">EXTREAL2:56</a></i>
<br/>.= 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="extreal2.html#T70" title="EXTREAL2:th.70">EXTREAL2:70</a></i>
<br/>.= 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="extreal2.html#T56" title="EXTREAL2:th.56">EXTREAL2:56</a></i>
;<br/>
<b>hence </b><a NAME="E12:31_2"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="#E8:31_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:31"/><b>then </b>
( <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2 is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2 is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> )
 
<b>by </b><i><a class="txt" href="#E8:31"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/>
<a NAME="E11:31"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
( <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T12" title="MESFUNC2:th.12">MESFUNC2:12</a></i>;<br/>
<a NAME="E12:31"/><i><font color="Green" title="E28">A16</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>proof </b></a><div class="add">

<a NAME="E1:31_3"/><i><font color="Green" title="E29">A17</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E11:31"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="mesfunc2.html#T2" title="MESFUNC2:th.2">MESFUNC2:2</a></i>;<br/>
<a NAME="E2:31_3"/>
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">X</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"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<b>assume </b><a NAME="E1:31_3_1"/><i><font color="Green" title="E30">A18</font></i>: 
<font color="Maroon" title="c6">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"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E2:31_3_1"/><b>then </b><i><font color="Green" title="E31">A19</font></i>: 
( <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> &amp; <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> &amp; <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> &amp; <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> )
 
<b>by </b><i><a class="txt" href="#E2:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E6:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a>, <a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E3:31_3_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E6:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="#E7:31"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="#E1:31_3_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E4:31_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="mesfunc7.html#K5" title="MESFUNC7:func.5">|^</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="mesfunc7.html#K5" title="MESFUNC7:func.5">|^</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E6:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="#E1:31_3_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="mesfunc7.html#D5" target="_self" title="MESFUNC7:def.5">Def5</a></i>;<br/>
<a NAME="E5:31_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="mesfunc7.html#K5" title="MESFUNC7:func.5">|^</a> 1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="mesfunc7.html#K5" title="MESFUNC7:func.5">|^</a> 1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc7.html#T10" target="_self" title="MESFUNC7:th.10">Th10</a></i>;<br/>
<a NAME="E6:31_3_1"/><b>then </b><i><font color="Green" title="E32">A20</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E2:31_3_1"><i><font color="Green" title="E31">A19</font></i></a>, <a class="ref" href="mesfunc7.html#T9" target="_self" title="MESFUNC7:th.9">Th9</a></i>;<br/>
<a NAME="E7:31_3_1"/>
( <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E5:31"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="#E6:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="#E1:31_3_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">c1</font> = <font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c8">c2</font> = <font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="extreal2.html#T102" title="EXTREAL2:th.102">EXTREAL2:102</a></i>;<br/>
<a NAME="E9:31_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font> )
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a>, <a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<a NAME="E10:31_3_1"/><b>then </b>
( <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span> &amp; <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="extreal2.html#T49" title="EXTREAL2:th.49">EXTREAL2:49</a></i>;<br/>
<a NAME="E11:31_3_1"/><b>then </b>
( <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> &amp; <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/>
<a NAME="E12:31_3_1"/><b>then </b><i><font color="Green" title="E33">A21</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E6:31_3_1"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/>
<a NAME="E13:31_3_1"/>
( <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
;<br/>
<a NAME="E14:31_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="complex1.html#T161" title="COMPLEX1:th.161">COMPLEX1:161</a></i>;<br/>
<a NAME="E15:31_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">c1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c2</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">c1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c2</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> )
 
;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">c1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c2</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">c1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">c1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c2</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E12:31_3_1"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>
<br/>.= 
<font color="Maroon" title="c7">c1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="#E1:31_3_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E17:31_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E6:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="#E7:31"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="#E1:31_3"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="#E1:31_3_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_3"/>
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E6:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="#E7:31"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="#E1:31_3"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:31"/>
( <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E5:31"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="mesfunc2.html#T7" title="MESFUNC2:th.7">MESFUNC2:7</a>, <a class="ref" href="mesfunc2.html#T13" title="MESFUNC2:th.13">MESFUNC2:13</a></i>;<br/>
<a NAME="E14:31"/><b>then </b>
( <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2 <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> &amp; <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2 <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="mesfunc7.html#T14" target="_self" title="MESFUNC7:th.14">Th14</a></i>;<br/>
<a NAME="E15:31"/><b>then </b>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> 2</span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="mesfunc5.html#T55" title="MESFUNC5:th.55">MESFUNC5:55</a></i>;<br/>
<b>hence </b><a NAME="E16:31"/>
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E7:31"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="#E11:31"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="#E12:31"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="mesfunc2.html#T13" title="MESFUNC2:th.13">MESFUNC2:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
