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

<b>assume </b><a NAME="E1:4"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">e</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> holds <br/><font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">e</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <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#K1">+infty</a>  &amp; <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="4_1_1"><b>suppose </b></a><a NAME="E1:4_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_1_1_2"/><i><font color="Green">E38</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="E2:4_1_1_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:4_1_1_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>suppose </b></a><a NAME="E1:4_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">y</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#K2">-infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1_1"><b>suppose </b></a><a NAME="E1:4_1_2_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:4_1_2_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:4_1_2_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1_2"><b>suppose </b></a><a NAME="E1:4_1_2_1_2"/><i><font color="Green">E41</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> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1_2_1_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><b>assume </b><a NAME="E2:4_1_2_1_2_1"/><i><font color="Green">E43</font></i>: 
not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E3:4_1_2_1_2_1"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a>, <a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/><a NAME="E4:4_1_2_1_2_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><a NAME="E5:4_1_2_1_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T41">EXTREAL1:41</a></i>;<br/><b>reconsider </b><font color="Maroon">x1</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/><b>reconsider </b><font color="Maroon">y1</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="E8:4_1_2_1_2_1"/><i><font color="Green">E47</font></i>: 
<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">x1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E9:4_1_2_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y1</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:4_1_2_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T11">EXTREAL1:11</a></i>;<br/><a NAME="E11:4_1_2_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T11">EXTREAL1:11</a>, </i>;<br/><a NAME="E12:4_1_2_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T9">EXTREAL1:9</a></i>;<br/><b>hence </b><a NAME="E13:4_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:4_1_2_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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