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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E38">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> or <font color="Maroon" title="c2">y</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">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" title="c2">y</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> or <font color="Maroon" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 </span><b>by </b><i><a class="txt" href="extreal2.html#E1:57"><i><font color="Green" title="E38">A1</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" title="E39">A2</font></i>: 
<font color="Maroon" title="c2">y</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 ;<br/><div class="add"><a NAME="E2:57_1_1"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 ;<br/><b>reconsider </b><font color="Maroon" title="c3">b</font> = <font color="Maroon" title="c2">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="extreal2.html#E1:57_1_1"><i><font color="Green" title="E39">A2</font></i></a></i>;<br/><a NAME="E4:57_1_1"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T49" target="_self" title="EXTREAL2:th.49">Th49</a></i>;<br/><a NAME="E5:57_1_1"/><b>then </b><i><font color="Green" title="E41">A4</font></i>: 
( <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 ;<br/><a NAME="E6:57_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:57_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="measure6.html#T8" title="MEASURE6:th.8">MEASURE6:8</a></i>;<br/><a NAME="E7:57_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 <b>by </b><i><a class="txt" href="extreal2.html#E2:57_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="extreal2.html#T61" target="_self" title="EXTREAL2:th.61">Th61</a></i>;<br/><b>hence </b><a NAME="E8:57_1_1"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 <b>by </b><i><a class="txt" href="extreal2.html#E5:57_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="extreal2.html#T34" target="_self" title="EXTREAL2:th.34">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" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c1">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E3:57_1_2"/><i><font color="Green" title="E39">A5</font></i>: 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T49" target="_self" title="EXTREAL2:th.49">Th49</a></i>;<br/><a NAME="E4:57_1_2"/><b>then </b><i><font color="Green" title="E40">A6</font></i>: 
( <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-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" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or ( <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-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" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-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" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T67" target="_self" title="EXTREAL2:th.67">Th67</a></i>;<br/><a NAME="E3:57_1_2_1_1_1"/><b>then </b>
( <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> )
 <b>by </b><i><a class="txt" href="extreal2.html#E4:57_1_2"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="extreal2.html#T51" target="_self" title="EXTREAL2:th.51">Th51</a>, <a class="ref" href="supinf_2.html#T6" title="SUPINF_2:th.6">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" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</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" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c2">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><a NAME="E3:57_1_2_1_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><a NAME="E4:57_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E41">A7</font></i>: 
( <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">b</font> &amp; <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">b</font></span>)</span> )
 <b>by </b><i><a class="ref" href="extreal2.html#T49" target="_self" title="EXTREAL2:th.49">Th49</a></i>;<br/><a NAME="E5:57_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E42">A8</font></i>: 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal2.html#E3:57_1_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">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" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T145" title="COMPLEX1:th.145">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" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 <b>by </b><i><a class="txt" href="extreal2.html#E4:57_1_2_1_1_2"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="extreal2.html#E5:57_1_2_1_1_2"><i><font color="Green" title="E42">A8</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" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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