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<div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green">E19</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">x</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="63_1_1"><b>suppose </b></a><a NAME="E1:63_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:63_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E6:63_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E3:63_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i>, , <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E4:63_1_1"/>
<font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i>, , <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a></i>;<br/><a NAME="E5:63_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></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#T18" target="_self">Th18</a></i>;<br/><a NAME="E6:63_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T23" target="_self">Th23</a>, <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a></i>;<br/><a NAME="E7:63_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> 2 &amp;  <a href="measure6.html#K1">R_EAL</a> 2 <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/><a NAME="E8:63_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> 2 &amp;  <a href="measure6.html#K1">R_EAL</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><b>hence </b><a NAME="E9:63_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T20" target="_self">Th20</a>, <a class="ref" href="extreal2.html#T24" target="_self">Th24</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2"><b>suppose </b></a><a NAME="E1:63_1_2"/>
<font color="Maroon">x</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">a</font> = <font color="Maroon">x</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_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="E1:63_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">y</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="63_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_2_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:63_1_2_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i/>;<br/><a NAME="E3:63_1_2_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E2:63_1_2_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="extreal2.html#E6:63_1_2_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E4:63_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E3:63_1_2_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="extreal2.html#E6:63_1_2_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a></i>;<br/><a NAME="E5:63_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></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#T18" target="_self">Th18</a></i>;<br/><a NAME="E6:63_1_2_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T23" target="_self">Th23</a>, <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a></i>;<br/><a NAME="E7:63_1_2_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> 2 &amp;  <a href="measure6.html#K1">R_EAL</a> 2 <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/><a NAME="E8:63_1_2_1_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> 2 &amp;  <a href="measure6.html#K1">R_EAL</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><b>hence </b><a NAME="E9:63_1_2_1_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T20" target="_self">Th20</a>, <a class="ref" href="extreal2.html#T24" target="_self">Th24</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_2_1_1_2"/>
<font color="Maroon">y</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">b</font> = <font color="Maroon">y</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="E3:63_1_2_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E4:63_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E5:63_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</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">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:63_1_2_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E7:63_1_2_1_1_2"/><i><font color="Green">E53</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> 2 &amp;  <a href="measure6.html#K1">R_EAL</a> 2 <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/><a NAME="E8:63_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="extreal2.html#T20" target="_self">Th20</a>, <a class="ref" href="extreal2.html#T23" target="_self">Th23</a>, <a class="ref" href="extreal1.html#T32">EXTREAL1:32</a></i>;<br/><b>hence </b><a NAME="E9:63_1_2_1_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T159">COMPLEX1:159</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:63_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
