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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="E1:57"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is   <a href="real_1.html#NM1">Real</a> or <font color="Maroon">c<sub>2</sub></font> is   <a href="real_1.html#NM1">Real</a> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1"/>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="real_1.html#NM1">Real</a> or <font color="Maroon">c<sub>1</sub></font> is   <a href="real_1.html#NM1">Real</a> )
 </span><b>by </b><i><a class="txt" href="extreal2.html#E1:57"><i><font color="Green">E38</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1"><b>suppose </b></a><a NAME="E1:57_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is   <a href="real_1.html#NM1">Real</a>
 ;<br/><div class="add"><a NAME="E2:57_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal2.html#E1:57_1_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E4:57_1_1"/>
<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="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="extreal2.html#T49" target="_self">Th49</a></i>;<br/><a NAME="E5:57_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( <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#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &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#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><a NAME="E6:57_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:57_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="measure6.html#T8">MEASURE6:8</a></i>;<br/><a NAME="E7:57_1_1"/><b>then </b>
<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="xxreal_0.html#R1">&lt;=</a> <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#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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#E2:57_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="extreal2.html#T61" target="_self">Th61</a></i>;<br/><b>hence </b><a NAME="E8:57_1_1"/>
<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#K4">-</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> <a href="xxreal_0.html#R1">&lt;=</a> <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#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="txt" href="extreal2.html#E5:57_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="extreal2.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2"><b>suppose </b></a><a NAME="E1:57_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="real_1.html#NM1">Real</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E3:57_1_2"/><i><font color="Green">E39</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="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="extreal2.html#T49" target="_self">Th49</a></i>;<br/><a NAME="E4:57_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#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <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#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_2_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:57_1_2_1_1_1"/><b>then </b>
<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_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T67" target="_self">Th67</a></i>;<br/><a NAME="E3:57_1_2_1_1_1"/><b>then </b>
( <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#K4">-</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> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <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#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> )
 <b>by </b><i><a class="txt" href="extreal2.html#E4:57_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="extreal2.html#T51" target="_self">Th51</a>, <a class="ref" href="supinf_2.html#T6">SUPINF_2:6</a></i>;<br/><b>hence </b><a NAME="E4:57_1_2_1_1_1"/>
<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#K4">-</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> <a href="xxreal_0.html#R1">&lt;=</a> <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#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_2_1_1_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E3:57_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E4:57_1_2_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
( <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="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font> &amp; <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#K4">-</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>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="extreal2.html#T49" target="_self">Th49</a></i>;<br/><a NAME="E5:57_1_2_1_1_2"/><b>then </b><i><font color="Green">E42</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="supinf_2.html#K4">-</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal2.html#E3:57_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E6:57_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T145">COMPLEX1:145</a></i>;<br/><b>hence </b><a NAME="E7:57_1_2_1_1_2"/>
<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#K4">-</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> <a href="xxreal_0.html#R1">&lt;=</a> <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#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="txt" href="extreal2.html#E4:57_1_2_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="extreal2.html#E5:57_1_2_1_1_2"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:57_1_2"/>
<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#K4">-</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> <a href="xxreal_0.html#R1">&lt;=</a> <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#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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