<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font> &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  implies <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:133"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">and </span><br/><a NAME="E2:133"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">and </span><br/><a NAME="E3:133"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font></span><br/>

<div><span class="lab"><font color="Green" title="E70">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:133_1"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a>, <a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></span>;<br/><span class="kw">assume </span><a NAME="E2:133_1"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:133_1_1"/>
( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_1"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:133_1_1_1"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:133_1_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E3:133"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c3">z</font> as   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_1_1_1"><span class="lab"><font color="Green" title="E73">A7</font></span></a></span>;<br/><a NAME="E4:133_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">y</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E5:133_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:133_1_1_2"/><span class="lab"><font color="Green" title="E73">A8</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:133_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="xxreal_3.html#D5" target="_self" title="XXREAL_3:def.5">Def5</a></span>;<br/><a NAME="E3:133_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E3:133"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E2:133_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="xxreal_3.html#T69" target="_self" title="XXREAL_3:th.69">Th80</a></span>;<br/><span class="kw">hence </span><a NAME="E4:133_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E2:133_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="txt" href="#E1:133_1_1_2"><span class="lab"><font color="Green" title="E73">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:133_1_1_3"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:133_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="xxreal_3.html#D5" target="_self" title="XXREAL_3:def.5">Def5</a></span>;<br/><a NAME="E3:133_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E3:133"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E2:133_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="xxreal_3.html#T70" target="_self" title="XXREAL_3:th.70">Th81</a></span>;<br/><span class="kw">hence </span><a NAME="E4:133_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E2:133_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="txt" href="#E1:133_1_1_3"><span class="lab"><font color="Green" title="E73">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E5:133"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="xxreal_3.html#T71" target="_self" title="XXREAL_3:th.71">Th82</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:133"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:133"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
