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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">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> ;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E31">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">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="c3">g</font></span>)</span> holds <br/>( <font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> <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>  )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c4">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="c2">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c3">g</font></span>)</span>
 ;<br/><a NAME="E2:33_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/><a NAME="E3:33_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">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="c3">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E4:33_1"/><b>then </b><i><font color="Green" title="E33">A3</font></i>: 
( <font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">x</font> <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="mesfunc5.html#E1:33"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E1:33_1"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/><a NAME="E5:33_1"/><b>then </b>
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">x</font> <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="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:33_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">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="c3">g</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:33_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="extreal1.html#T40" title="EXTREAL1:th.40">EXTREAL1:40</a></i>;<br/><b>hence </b><a NAME="E7:33_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:33_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33"/>
<font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c3">g</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>;<br/>


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