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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> implies  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">x</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:68"/><span class="lab"><font color="Green" title="E44">A1</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="c2">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">x</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_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="E1:68_1"/>
( (  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</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>  &amp;  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <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>  ) or  not  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</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  not  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <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>;<br/> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xxreal_3.html#D1" target="_self" title="XXREAL_3:def.1">XXREAL_3:def 1</a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1"><span class="kw">case </span></a><span class="kw">that </span><a NAME="E1:68_1_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</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> 
 <span class="kw">and </span><br/><a NAME="E2:68_1_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <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">  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <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>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">q</font> )</span><br/><div class="add"><a NAME="E3:68_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68_1_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="xxreal_3.html#E15"><span class="lab"><font color="Green" title="E13">Lm3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E5:68_1_1"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E6:68_1_1"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">a</font>
 <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="E7:68_1_1"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="xxreal_3.html#E15"><span class="lab"><font color="Green" title="E13">Lm3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">b</font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E9:68_1_1"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E10:68_1_1"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">b</font>
 <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="E11:68_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E2:68_1_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="xxreal_3.html#E15"><span class="lab"><font color="Green" title="E13">Lm3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c4">b</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="#E9:68_1_1"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E5:68_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">mb</font> =  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c8">ma</font> =  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c5">a</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="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c7">mb</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q</font> being   <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>  st <br/>( <font color="Maroon" title="c7">mb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c7">mb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c8">ma</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="c7">mb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c8">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c7">mb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">ma</font> )</span><br/><span class="kw">thus </span><a NAME="E14:68_1_1"/>
( <font color="Maroon" title="c7">mb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c8">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:68_1_1"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="txt" href="#E6:68_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a></span>; <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="c7">mb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">ma</font></span><br/><span class="kw">thus </span><a NAME="E15:68_1_1"/>
<font color="Maroon" title="c7">mb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E9:68_1_1"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E5:68_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</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="68_1_2"><span class="kw">case </span></a><a NAME="E1:68_1_2"/>
(  not  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</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  not  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <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"> (  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</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>  or  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <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>  )</span><br/><div class="add"><a NAME="E2:68_1_2"/><span class="kw">then </span>
(  not <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  not <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 class="kw">by</span> <span class="lab"><a class="txt" href="xxreal_3.html#E15"><span class="lab"><font color="Green" title="E13">Lm3</font></span></a></span>;<br/><a NAME="E3:68_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#K2" title="XXREAL_0:func.2">-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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D1" target="_self" title="XXREAL_3:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:68_1_2"/>
(  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</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>  or  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <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>  )
 <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>; <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>

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