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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><b>that </b><br/><a NAME="E1:37"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 ;<br/>

<a NAME="E3:37"/><i><font color="Green">E37</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="txt" href="mesfunc5.html#E1:37"><i><font color="Green">E35</font></i></a>, <a class="txt" href="mesfunc5.html#E2:37"><i><font color="Green">E36</font></i></a>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E4:37"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</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="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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</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="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="mesfunc2.html#D2">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E5:37"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</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> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</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="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</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> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</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="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:37"><i><font color="Green">E37</font></i></a>, <a class="ref" href="mesfunc2.html#T14">MESFUNC2:14</a>, <a class="ref" href="mesfunc2.html#T15">MESFUNC2:15</a></i>;<br/>
<a NAME="E6:37"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="mesfunc2.html#K1">max+</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> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp;  <a href="mesfunc2.html#K2">max-</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> 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/>
<a NAME="E7:37"/><i><font color="Green">E40</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E8:37"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T14">MESFUNC2:14</a>, <a class="ref" href="mesfunc2.html#T15">MESFUNC2:15</a></i>;<br/>
<a NAME="E9:37"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp;  <a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp;  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>3</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp;  <a href="mesfunc2.html#K2">max-</a> <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/>
<a NAME="E10:37"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:37"><i><font color="Green">E39</font></i></a>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E11:37"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</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>2</sub></font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</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>2</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:37"><i><font color="Green">E40</font></i></a>, <a class="txt" href="mesfunc5.html#E4:37"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<b>hence </b><a NAME="E12:37"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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> )
 ;<br/>

<a NAME="E13:37"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> implies 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> implies 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:37_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</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>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:37"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mesfunc5.html#E7:37"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:37_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</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="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</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#E3:37"><i><font color="Green">E37</font></i></a>, <a class="txt" href="mesfunc5.html#E11:37"><i><font color="Green">E43</font></i></a>, <a class="txt" href="mesfunc5.html#E1:37_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="mesfunc2.html#T14">MESFUNC2:14</a>, <a class="ref" href="mesfunc2.html#T15">MESFUNC2:15</a></i>;<br/><a NAME="E4:37_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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="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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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="E5:37_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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#E1:37_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:37_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:37_1"/><b>then </b>
<font color="Maroon">c<sub>4</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>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:37"><i><font color="Green">E40</font></i></a>, <a class="txt" href="mesfunc5.html#E10:37"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:37_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</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="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</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#E3:37"><i><font color="Green">E37</font></i></a>, <a class="txt" href="mesfunc5.html#E11:37"><i><font color="Green">E43</font></i></a>, <a class="txt" href="mesfunc5.html#E2:37_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="mesfunc2.html#T14">MESFUNC2:14</a>, <a class="ref" href="mesfunc2.html#T15">MESFUNC2:15</a></i>;<br/>
<a NAME="E5:37_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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="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"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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="E6:37_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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#E2:37_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:37"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<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="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
;<br/>
<a NAME="E15:37"/><b>then </b><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</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="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp; <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</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="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> 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>then </b> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<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="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<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>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:37"><i><font color="Green">E40</font></i></a>, <a class="txt" href="mesfunc5.html#E9:37"><i><font color="Green">E41</font></i></a>, <a class="txt" href="mesfunc5.html#E11:37"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a></i>
<br/>.= 
<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"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</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>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/>
<b>hence </b><a NAME="E17:37"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<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="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>
 ;<br/>

<a NAME="E18:37"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<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="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:37"><i><font color="Green">E40</font></i></a>, <a class="txt" href="mesfunc5.html#E11:37"><i><font color="Green">E43</font></i></a>, <a class="txt" href="mesfunc5.html#E9:37"><i><font color="Green">E41</font></i></a>, <a class="txt" href="mesfunc5.html#E15:37"><i><font color="Green">E45</font></i></a>, <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E19:37"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<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="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</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>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<b>hence </b><a NAME="E20:37"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<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="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>
 ;<br/>

<b>thus </b><a NAME="E21:37"/>
( <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</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="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</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="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E13:37"><i><font color="Green">E44</font></i></a>, <a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/>


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