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

<b>assume </b><a NAME="E1:6"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><b>suppose </b></a><a NAME="E1:6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1"/>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2"><b>suppose </b></a><a NAME="E1:6_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:6_1_2"/><i><font color="Green">E22</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K7">}</a></span> ) &amp; ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K7">}</a></span> ) )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:6_1_2"/><i><font color="Green">E25</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or  not <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_1.html#T23">SUPINF_1:23</a></i>;<br/><a NAME="E4:6_1_2"/><i><font color="Green">E26</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <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="E5:6_1_2"/><i><font color="Green">E33</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i>, , <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/><a NAME="E6:6_1_2"/><i><font color="Green">E34</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <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="xxreal_0.html#T7">XXREAL_0:7</a></i>;<br/><a NAME="E7:6_1_2"/><i><font color="Green">E35</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_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="E1:6_1_2_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">y</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#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <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#K1">+infty</a>  ) )
 </span><b>by </b><i>, , , , , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font>, <font color="Maroon">c</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/><a NAME="E3:6_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E4:6_1_2_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font>, <font color="Maroon">r</font> = <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c</font>, <font color="Maroon">q</font> = <font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E7:6_1_2_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E8:6_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E9:6_1_2_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font>, <font color="Maroon">r</font> = <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font>, <font color="Maroon">q</font> = <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E12:6_1_2_1_1_1"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><b>hence </b><a NAME="E13:6_1_2_1_1_1"/>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:6_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#T6">SUPINF_2:6</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_1_1_2"/>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</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="6_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:6_1_2_1_1_3"/>
( <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#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><a NAME="E2:6_1_2_1_1_3"/><b>then </b>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_1_1_3"/>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:6_1_2_1_1_4"/>
( <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#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:6_1_2_1_1_4"/><b>then </b>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#T6">SUPINF_2:6</a>, <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_1_1_4"/>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</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><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:6_1_2"/>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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