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<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="Red">::  thesis: </font></span></a><span class="hide"> ( ( <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>  &amp; <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>  implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font> ) &amp; ( <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>  &amp; <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>  implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font> ) &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">y</font> implies <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:77"/><span class="lab"><font color="Green" title="E48">A1</font></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>  &amp; <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>  implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font> )
 <span class="kw">and </span><br/><a NAME="E2:77"/><span class="lab"><font color="Green" title="E49">A2</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>  &amp; <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>  implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font> )
 <span class="kw">and </span><br/><a NAME="E3:77"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_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="E1:77_1"/>
( <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>  or <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>  )
 </span><span class="kw">by</span> <span class="lab"><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="77_1_1"><span class="kw">suppose </span></a><a NAME="E1:77_1_1"/><span class="lab"><font color="Green" title="E51">A4</font></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_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="E1:77_1_1_1"/>
( <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>  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>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E1:77_1_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="xxreal_3.html#E22"><span class="lab"><font color="Green" title="E20">Lm8</font></span></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:77_1_1_1_1"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:77_1_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="77_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:77_1_1_1_2"/>
<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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:77_1_1_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E1:77_1_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="xxreal_3.html#T7" target="_self" title="XXREAL_3:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2"><span class="kw">suppose </span></a><a NAME="E1:77_1_2"/><span class="lab"><font color="Green" title="E51">A5</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_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="E1:77_1_2_1"/>
( <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>  or  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:77_1_2_1_1"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:77_1_2_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E1:77_1_2"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="xxreal_3.html#T7" target="_self" title="XXREAL_3:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="77_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:77_1_2_1_2"/>
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><a NAME="E2:77_1_2_1_2"/><span class="kw">then </span>
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">y</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D3" target="_self" title="XXREAL_3:def.3">Def3</a></span>;<br/><a NAME="E3:77_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <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="#E1:77_1_2"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:77_1_2_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_3"><span class="kw">suppose </span></a><a NAME="E1:77_1_3"/><span class="lab"><font color="Green" title="E51">A6</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_3_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="E1:77_1_3_1"/>
( <font color="Maroon" title="c3">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <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="c3">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <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>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E1:77_1_3"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T10" title="XXREAL_0:th.10">XXREAL_0:10</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:77_1_3_1_1"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_3_1_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="E1:77_1_3_1_1_1"/>
( <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="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>  or ( <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>  &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>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1_3_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="xxreal_3.html#T20" target="_self" title="XXREAL_3:th.20">Th20</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:77_1_3_1_1_1_1"/>
<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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><a NAME="E2:77_1_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <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="#E1:77_1_3"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a>, <a class="ref" href="xxreal_3.html#T5" target="_self" title="XXREAL_3:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:77_1_3_1_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="77_1_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:77_1_3_1_1_1_2"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:77_1_3_1_1_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="77_1_3_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:77_1_3_1_1_1_3"/>
( <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>  &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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c6">c</font> = <font color="Maroon" title="c3">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1_3"><span class="lab"><font color="Green" title="E51">A6</font></span></a></span>;<br/><a NAME="E3:77_1_3_1_1_1_3"/>
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77"><span class="lab"><font color="Green" title="E50">A3</font></span></a></span>;<br/><a NAME="E4:77_1_3_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/><span class="kw">hence </span><a NAME="E5:77_1_3_1_1_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:77_1_3_1_2"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_3_1_2_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="E1:77_1_3_1_2_1"/>
( <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>  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>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1_3_1_2"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="xxreal_3.html#E22"><span class="lab"><font color="Green" title="E20">Lm8</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_3_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:77_1_3_1_2_1_1"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:77_1_3_1_2_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="77_1_3_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:77_1_3_1_2_1_2"/>
<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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><a NAME="E2:77_1_3_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <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="#E1:77_1_3"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a>, <a class="ref" href="xxreal_3.html#T5" target="_self" title="XXREAL_3:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:77_1_3_1_2_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
