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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>, <font color="Maroon">c<sub>4</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="E1:36"/><i><font color="Green">E34</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> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green">E35</font></i>: 
<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="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E3:36"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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#K13">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<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></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>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>
<b>hence </b><a NAME="E4:36"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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#K13">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>

<b>thus </b><a NAME="E5:36"/>
<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#K13">+</a> <font color="Maroon">c<sub>4</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:36"><i><font color="Green">E35</font></i></a>, <a class="txt" href="mesfunc5.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font></span>)</span>
 ;<br/><a NAME="E2:36_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<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></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>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E3:36_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<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#K13">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<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>5</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:36_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="mesfunc5.html#E3:36"><i><font color="Green">E36</font></i></a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:36_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>
;<br/></div>
<b>end;</b></div>


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