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<div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<a NAME="E1:91"/><i><font color="Green" title="E67">A1</font></i>: 
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c1">x</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="c2">y</font> implies <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:91_1"/><i><font color="Green" title="E68">A2</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c1">x</font>
 <b>and </b><br/><a NAME="E2:91_1"/><i><font color="Green" title="E69">A3</font></i>: 
<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>
 ;<br/>

<a NAME="E3:91_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="complex1.html#E1:91_1"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E4:91_1"/><b>then </b>
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 implies <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> )
 
<b>by </b><i><a class="txt" href="complex1.html#E174"><i><font color="Green" title="E57">Lm23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:91_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="complex1.html#E2:91_1"><i><font color="Green" title="E69">A3</font></i></a>, <a class="ref" href="complex1.html#T129" target="_self" title="COMPLEX1:th.129">Th129</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:91"/>
( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> implies (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c1">x</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="c2">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_2"/><i><font color="Green" title="E68">A4</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 ;<br/>

<a NAME="E2:91_2"/><b>then </b><i><font color="Green" title="E69">A5</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132" target="_self" title="COMPLEX1:th.132">Th132</a></i>;<br/>
<a NAME="E3:91_2"/><i><font color="Green" title="E70">A6</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> implies (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c1">x</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="c2">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_2_1"/><i><font color="Green" title="E71">A7</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 ;<br/>

<a NAME="E2:91_2_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 0
 
<b>by </b><i><a class="txt" href="complex1.html#E2:91_2"><i><font color="Green" title="E69">A5</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E3:91_2_1"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c1">x</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="c2">y</font> )
 <b>by </b><i><a class="txt" href="complex1.html#E1:91_2"><i><font color="Green" title="E68">A4</font></i></a>, <a class="txt" href="complex1.html#E1:91_2_1"><i><font color="Green" title="E71">A7</font></i></a>, <a class="ref" href="complex1.html#T129" target="_self" title="COMPLEX1:th.129">Th129</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:91_2"/>
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 0 implies (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c1">x</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="c2">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_2_2"/><i><font color="Green" title="E71">A8</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 0
 ;<br/>

<a NAME="E2:91_2_2"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <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>
 
<b>by </b><i><a class="txt" href="complex1.html#E1:91_2"><i><font color="Green" title="E68">A4</font></i></a>, <a class="txt" href="complex1.html#E174"><i><font color="Green" title="E57">Lm23</font></i></a></i>;<br/>
<a NAME="E3:91_2_2"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E4:91_2_2"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c1">x</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="c2">y</font> )
 <b>by </b><i><a class="txt" href="complex1.html#E1:91_2"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="complex1.html#T132" target="_self" title="COMPLEX1:th.132">Th132</a>, <a class="txt" href="complex1.html#E1:91_2_2"><i><font color="Green" title="E71">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:91_2"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c1">x</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="c2">y</font> )
 <b>by </b><i><a class="txt" href="complex1.html#E2:91_2"><i><font color="Green" title="E69">A5</font></i></a>, <a class="txt" href="complex1.html#E3:91_2"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:91"/>
( (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c1">x</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="c2">y</font> ) iff <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> )
 <b>by </b><i><a class="txt" href="complex1.html#E1:91"><i><font color="Green" title="E67">A1</font></i></a></i>;<br/>


</div>
