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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:62"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1"/>
( ( ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> ) ) or ( ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) ) )
 </span><b>by </b><i><a class="txt" href="extreal2.html#E1:62"><i><font color="Green">E38</font></i></a>, <a class="ref" href="extreal2.html#T44" target="_self">Th44</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><b>suppose </b></a><a NAME="E1:62_1_1"/><i><font color="Green">E39</font></i>: 
( ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/><div class="add"><a NAME="E2:62_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>;<br/><a NAME="E3:62_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:62_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="extreal2.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E4:62_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="txt" href="extreal2.html#E2:62_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_2"><b>suppose </b></a><a NAME="E1:62_1_2"/><i><font color="Green">E39</font></i>: 
( ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) )
 ;<br/><div class="add"><a NAME="E2:62_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
( <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="extreal2.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E3:62_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="extreal2.html#E1:62_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="extreal2.html#T9" target="_self">Th9</a></i>;<br/><b>then </b><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> = 
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="extreal2.html#T55" target="_self">Th55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="extreal2.html#T14" target="_self">Th14</a></i>
;<br/><b>hence </b><a NAME="E5:62_1_2"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="txt" href="extreal2.html#E2:62_1_2"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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