<?xml version="1.0"?>
<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>let </b><font color="Maroon">z</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>





<b>assume </b><a NAME="E1:22"/>
( <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">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/>

<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="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1"/>
( <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_2.html#K1">0.</a>  or ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">z</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#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 </span><b>by </b><i><a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1"><b>suppose </b></a><a NAME="E1:22_1_1"/><i><font color="Green">E19</font></i>: 
<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_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:22_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:22_1_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1_1"/>
( <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#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <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="supinf_2.html#T2">SUPINF_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<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="E2:22_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1_1_1_1_1_1"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E2:22_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, , <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_1_1_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, , <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:22_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</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="extreal1.html#T8" target="_self">Th8</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">a'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:22_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">a'</font> )
 <b>by </b><i><a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, <a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/><a NAME="E7:22_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, <a class="txt" href="extreal1.html#E2:22_1_1_1"><i><font color="Green">E25</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:22_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="txt" href="extreal1.html#E2:22_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, , <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:22_1_1_1_1_3"/><i><font color="Green">E33</font></i>: 
<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="E2:22_1_1_1_1_3"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/><b>hence </b><a NAME="E3:22_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <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><b>then </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="txt" href="extreal1.html#E1:22_1_1"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E4:22_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2"><b>suppose </b></a><a NAME="E1:22_1_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:22_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><i><font color="Green">E35</font></i>: <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span>

;<br/><a NAME="E5:22_1_2"/>
( <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:22_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_3"><b>suppose </b></a><a NAME="E1:22_1_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">z</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="hs2(this)" href="javascript:()" title="22_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_3_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_3_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_3_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:22_1_3_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_3_1_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">z</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="22_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_3_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:22_1_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><a NAME="E3:22_1_3_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <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/><b>hence </b><a NAME="E4:22_1_3_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, , <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_3_1_1_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:22_1_3_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E4:22_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E5:22_1_3_1_1_1_1_1_2"/>
<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> 
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E6:22_1_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E7:22_1_3_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, <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><b>hence </b><a NAME="E5:22_1_3_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_3_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_3_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:22_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:22_1_3_1_1_3"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_3_1_1_3"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E3:22_1_3_1_1_3"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_3_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_3_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_3_1_1_3_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">z</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="22_1_3_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_3_1_1_3_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:22_1_3_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><a NAME="E3:22_1_3_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:22_1_3_1_1_3_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, , <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_3_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_3_1_1_3_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:22_1_3_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E4:22_1_3_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E5:22_1_3_1_1_3_1_1_2"/>
<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> 
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E6:22_1_3_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E7:22_1_3_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, <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><b>hence </b><a NAME="E5:22_1_3_1_1_3"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:22_1_3"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_4"><b>suppose </b></a><a NAME="E1:22_1_4"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">z</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="22_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_4_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_4_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_4_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:22_1_4_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_4_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_4_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_4_1_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">z</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="22_1_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_4_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:22_1_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><a NAME="E3:22_1_4_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:22_1_4_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, , <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_4_1_1_1_1_1_2"/>
<font color="Maroon">z</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</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:22_1_4_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E4:22_1_4_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E5:22_1_4_1_1_1_1_1_2"/>
<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#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E6:22_1_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E7:22_1_4_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, <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><b>hence </b><a NAME="E5:22_1_4_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_4_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_4_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:22_1_4_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:22_1_4_1_1_3"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_4_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E3:22_1_4_1_1_3"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_4_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_4_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_4_1_1_3_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">z</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="22_1_4_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_4_1_1_3_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:22_1_4_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><a NAME="E3:22_1_4_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <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/><b>hence </b><a NAME="E4:22_1_4_1_1_3_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, , <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_4_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_4_1_1_3_1_1_2"/>
<font color="Maroon">z</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</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:22_1_4_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E4:22_1_4_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E5:22_1_4_1_1_3_1_1_2"/>
<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#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E6:22_1_4_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E7:22_1_4_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, <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><b>hence </b><a NAME="E5:22_1_4_1_1_3"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:22_1_4"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5"><b>suppose </b></a><a NAME="E1:22_1_5"/><i><font color="Green">E65</font></i>: 
( <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="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="hs2(this)" href="javascript:()" title="22_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_5_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_5_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_5_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:22_1_5_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_5_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_5_1_1_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#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="22_1_5_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_5_1_1_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"><a NAME="E2:22_1_5_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><a NAME="E3:22_1_5_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <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/><b>hence </b><a NAME="E4:22_1_5_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, , <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_5_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+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="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:22_1_5_1_1_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E4:22_1_5_1_1_1_1_1_2"/>
<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> 
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E5:22_1_5_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:22_1_5_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:22_1_5_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_5_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_5_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:22_1_5_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:22_1_5_1_1_3"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_5_1_1_3"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E3:22_1_5_1_1_3"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_5_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_5_1_1_3_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#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="22_1_5_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_5_1_1_3_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"><a NAME="E2:22_1_5_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><a NAME="E3:22_1_5_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:22_1_5_1_1_3_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, , <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_5_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_5_1_1_3_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+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="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:22_1_5_1_1_3_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E4:22_1_5_1_1_3_1_1_2"/>
<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> 
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E5:22_1_5_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:22_1_5_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:22_1_5_1_1_3"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:22_1_5"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_6"><b>suppose </b></a><a NAME="E1:22_1_6"/><i><font color="Green">E72</font></i>: 
( <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="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="22_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_6_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_6_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_6_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:22_1_6_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_6_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_6_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_6_1_1_1_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="22_1_6_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_6_1_1_1_1_1_1"/>
<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="E2:22_1_6_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><a NAME="E3:22_1_6_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:22_1_6_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, , <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_6_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_6_1_1_1_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="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:22_1_6_1_1_1_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E4:22_1_6_1_1_1_1_1_2"/>
<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#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E5:22_1_6_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:22_1_6_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:22_1_6_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_6_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_6_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:22_1_6_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:22_1_6_1_1_3"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_6_1_1_3"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E3:22_1_6_1_1_3"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_6_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_6_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_6_1_1_3_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="22_1_6_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_6_1_1_3_1_1_1"/>
<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="E2:22_1_6_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</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="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><a NAME="E3:22_1_6_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <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/><b>hence </b><a NAME="E4:22_1_6_1_1_3_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, , <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_6_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_6_1_1_3_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="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:22_1_6_1_1_3_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E4:22_1_6_1_1_3_1_1_2"/>
<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#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E5:22_1_6_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:22_1_6_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:22_1_6_1_1_3"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:22_1_6"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
