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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 ;<br/>

<a NAME="E2:35"/><b>then </b>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/>
<a NAME="E3:35"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E4:35"/><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E2:35_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:35"><i><font color="Green">E33</font></i></a>, <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/><a NAME="E3:35_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T7">EXTREAL2:7</a></i>;<br/><b>hence </b><a NAME="E4:35_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:35_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="mesfunc5.html#E4:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/>


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