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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</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/>





<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c5">y1</font> = <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c6">z1</font> = <font color="Maroon" title="c3">c</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>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:4"/><i><font color="Green" title="E7">A2</font></i>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> )
 
<b>by </b><i><a class="txt" href="xreal_1.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1"/>
( ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ) or ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ) or ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ) or ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) or ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) or ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 </span><b>by </b><i><a class="txt" href="xreal_1.html#E1:4"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:4_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:4_1_1"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:4_1_1"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c7">a''</font>, <font color="Maroon" title="c8">b''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E5:4_1_1"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a''</font>
 <b>and </b><br/><a NAME="E6:4_1_1"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b''</font>
 <b>and </b><br/><a NAME="E7:4_1_1"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c7">a''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">b''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:4"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="xreal_1.html#E1:4_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">c'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E9:4_1_1"/><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font>
 <b>and </b><br/><a NAME="E10:4_1_1"/><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font>
 <b>and </b><br/><a NAME="E11:4_1_1"/><i><font color="Green" title="E16">A11</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">c'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:4_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c12">c''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E13:4_1_1"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b'</font>
 <b>and </b><br/><a NAME="E14:4_1_1"/><i><font color="Green" title="E18">A13</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c''</font>
 <b>and </b><br/><a NAME="E15:4_1_1"/><i><font color="Green" title="E19">A14</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:4_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><a NAME="E16:4_1_1"/>
<font color="Maroon" title="c9">a'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">b'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:4_1_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="xreal_1.html#E6:4_1_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="xreal_1.html#E7:4_1_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="xreal_1.html#E9:4_1_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="xreal_1.html#E10:4_1_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="xreal_1.html#E13:4_1_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="xreal_1.html#E14:4_1_1"><i><font color="Green" title="E18">A13</font></i></a>, <a class="ref" href="arytm_1.html#T7" title="ARYTM_1:th.7">ARYTM_1:7</a></i>;<br/><b>hence </b><a NAME="E17:4_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="xreal_1.html#E11:4_1_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="xreal_1.html#E15:4_1_1"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:4_1_2"/><i><font color="Green" title="E8">A15</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E2:4_1_2"/><i><font color="Green" title="E9">A16</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:4_1_2"/><i><font color="Green" title="E10">A17</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c8">c'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E5:4_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c7">a'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:4_1_2"/><i><font color="Green" title="E11">A18</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c'</font>
 <b>and </b><br/><a NAME="E7:4_1_2"/><i><font color="Green" title="E12">A19</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c7">a'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:4_1_2"><i><font color="Green" title="E8">A15</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_2"><i><font color="Green" title="E10">A17</font></i></a>, <a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">b'</font>, <font color="Maroon" title="c10">c''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E9:4_1_2"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font>
 <b>and </b><br/><a NAME="E10:4_1_2"/><i><font color="Green" title="E13">A20</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c''</font>
 <b>and </b><br/><a NAME="E11:4_1_2"/><i><font color="Green" title="E14">A21</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:4_1_2"><i><font color="Green" title="E9">A16</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_2"><i><font color="Green" title="E10">A17</font></i></a>, <a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_2_1_1"/>
( <font color="Maroon" title="c7">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">c'</font> or  not <font color="Maroon" title="c7">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">c'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_2_1_1_1"/>
<font color="Maroon" title="c7">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">c'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E15">A22</font></i>: 
<font color="Maroon" title="c8">c'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">a'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_2_1_1_1"/><i><font color="Green" title="E16">A23</font></i>: 
<font color="Maroon" title="c8">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">c'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#T11" title="ARYTM_1:th.11">ARYTM_1:11</a></i>;<br/><a NAME="E4:4_1_2_1_1_1"/>
<font color="Maroon" title="c8">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">b'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:4_1_2"><i><font color="Green" title="E11">A18</font></i></a>, <a class="txt" href="xreal_1.html#E10:4_1_2"><i><font color="Green" title="E13">A20</font></i></a>, <a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></i>;<br/><a NAME="E5:4_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">b'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4_1_2_1_1_1"><i><font color="Green" title="E16">A23</font></i></a>, <a class="ref" href="arytm_1.html#T3" title="ARYTM_1:th.3">ARYTM_1:3</a></i>;<br/><b>hence </b><a NAME="E6:4_1_2_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="xreal_1.html#E7:4_1_2"><i><font color="Green" title="E12">A19</font></i></a>, <a class="txt" href="xreal_1.html#E11:4_1_2"><i><font color="Green" title="E14">A21</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_2_1_1_1"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_2_1_1_2"/>
not <font color="Maroon" title="c7">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">c'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">c'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">c'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">c'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:4_1_2_1_1_2"/><b>then </b>
( not <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="xreal_1.html#E7:4_1_2"><i><font color="Green" title="E12">A19</font></i></a>, <a class="txt" href="xreal_1.html#E11:4_1_2"><i><font color="Green" title="E14">A21</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E5:4_1_2_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:4_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:4_1_3"/><i><font color="Green" title="E8">A24</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E2:4_1_3"/><i><font color="Green" title="E9">A25</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E3:4_1_3"/><i><font color="Green" title="E10">A26</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c7">a''</font>, <font color="Maroon" title="c8">b''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E5:4_1_3"/><i><font color="Green" title="E11">A27</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c7">a''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:4_1_3"/><i><font color="Green" title="E12">A28</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">b''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:4_1_3"/><i><font color="Green" title="E13">A29</font></i>: 
<font color="Maroon" title="c8">b''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">a''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:4"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="xreal_1.html#E1:4_1_3"><i><font color="Green" title="E8">A24</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_3"><i><font color="Green" title="E9">A25</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">c'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E9:4_1_3"/><i><font color="Green" title="E14">A30</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c9">a'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E10:4_1_3"/><i><font color="Green" title="E15">A31</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font>
 <b>and </b><br/><a NAME="E11:4_1_3"/><i><font color="Green" title="E16">A32</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:4_1_3"><i><font color="Green" title="E8">A24</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_3"><i><font color="Green" title="E10">A26</font></i></a>, <a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c12">c''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E13:4_1_3"/><i><font color="Green" title="E17">A33</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">b'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E14:4_1_3"/><i><font color="Green" title="E18">A34</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c''</font>
 <b>and </b><br/><a NAME="E15:4_1_3"/><i><font color="Green" title="E19">A35</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c11">b'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:4_1_3"><i><font color="Green" title="E9">A25</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_3"><i><font color="Green" title="E10">A26</font></i></a>, <a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><a NAME="E16:4_1_3"/><i><font color="Green" title="E20">A36</font></i>: 
<font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:4_1_3"><i><font color="Green" title="E11">A27</font></i></a>, <a class="txt" href="xreal_1.html#E9:4_1_3"><i><font color="Green" title="E14">A30</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E17:4_1_3"/><i><font color="Green" title="E21">A37</font></i>: 
<font color="Maroon" title="c11">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:4_1_3"><i><font color="Green" title="E12">A28</font></i></a>, <a class="txt" href="xreal_1.html#E13:4_1_3"><i><font color="Green" title="E17">A33</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_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="E1:4_1_3_1_1"/>
( <font color="Maroon" title="c9">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">c'</font> or  not <font color="Maroon" title="c9">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">c'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_3_1_1_1"/><i><font color="Green" title="E22">A38</font></i>: 
<font color="Maroon" title="c9">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">c'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E23">A39</font></i>: 
<font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_3_1_1_1"/>
<font color="Maroon" title="c11">b'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">c'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E7:4_1_3"><i><font color="Green" title="E13">A29</font></i></a>, <a class="txt" href="xreal_1.html#E16:4_1_3"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="xreal_1.html#E17:4_1_3"><i><font color="Green" title="E21">A37</font></i></a>, <a class="txt" href="xreal_1.html#E1:4_1_3_1_1_1"><i><font color="Green" title="E22">A38</font></i></a>, <a class="ref" href="arytm_1.html#T3" title="ARYTM_1:th.3">ARYTM_1:3</a></i>;<br/><a NAME="E4:4_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E24">A40</font></i>: 
<font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c11">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">b'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E5:4_1_3_1_1_1"/>
<font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c12">c''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">b'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E7:4_1_3"><i><font color="Green" title="E13">A29</font></i></a>, <a class="txt" href="xreal_1.html#E10:4_1_3"><i><font color="Green" title="E15">A31</font></i></a>, <a class="txt" href="xreal_1.html#E14:4_1_3"><i><font color="Green" title="E18">A34</font></i></a>, <a class="txt" href="xreal_1.html#E16:4_1_3"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="xreal_1.html#E17:4_1_3"><i><font color="Green" title="E21">A37</font></i></a>, <a class="ref" href="arytm_1.html#T16" title="ARYTM_1:th.16">ARYTM_1:16</a></i>;<br/><b>hence </b><a NAME="E6:4_1_3_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="xreal_1.html#E10:4_1_3"><i><font color="Green" title="E15">A31</font></i></a>, <a class="txt" href="xreal_1.html#E11:4_1_3"><i><font color="Green" title="E16">A32</font></i></a>, <a class="txt" href="xreal_1.html#E14:4_1_3"><i><font color="Green" title="E18">A34</font></i></a>, <a class="txt" href="xreal_1.html#E15:4_1_3"><i><font color="Green" title="E19">A35</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_3_1_1_1"><i><font color="Green" title="E23">A39</font></i></a>, <a class="txt" href="xreal_1.html#E4:4_1_3_1_1_1"><i><font color="Green" title="E24">A40</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_3_1_1_2"/>
not <font color="Maroon" title="c9">a'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">c'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E22">A41</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c9">a'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">c'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E11:4_1_3"><i><font color="Green" title="E16">A32</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E23">A42</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_3_1_1_2_1_1"/>
( <font color="Maroon" title="c11">b'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">c'</font> or  not <font color="Maroon" title="c11">b'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">c'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">b'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">c'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c11">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">b'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_3_1_1_2_1_1_1"/><b>then </b>
( not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="xreal_1.html#E10:4_1_3"><i><font color="Green" title="E15">A31</font></i></a>, <a class="txt" href="xreal_1.html#E14:4_1_3"><i><font color="Green" title="E18">A34</font></i></a>, <a class="txt" href="xreal_1.html#E15:4_1_3"><i><font color="Green" title="E19">A35</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_3_1_1_2"><i><font color="Green" title="E23">A42</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E4:4_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_3_1_1_2_1_1_2"/>
not <font color="Maroon" title="c11">b'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">c'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E24">A43</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c11">b'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">c'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E10:4_1_3"><i><font color="Green" title="E15">A31</font></i></a>, <a class="txt" href="xreal_1.html#E14:4_1_3"><i><font color="Green" title="E18">A34</font></i></a>, <a class="txt" href="xreal_1.html#E15:4_1_3"><i><font color="Green" title="E19">A35</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E25">A44</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:4_1_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c11">b'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">a'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">c'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E7:4_1_3"><i><font color="Green" title="E13">A29</font></i></a>, <a class="txt" href="xreal_1.html#E16:4_1_3"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="xreal_1.html#E17:4_1_3"><i><font color="Green" title="E21">A37</font></i></a>, <a class="ref" href="arytm_1.html#T17" title="ARYTM_1:th.17">ARYTM_1:17</a></i>;<br/><b>hence </b><a NAME="E5:4_1_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_3_1_1_2"><i><font color="Green" title="E22">A41</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_3_1_1_2"><i><font color="Green" title="E23">A42</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_3_1_1_2_1_1_2"><i><font color="Green" title="E24">A43</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_3_1_1_2_1_1_2"><i><font color="Green" title="E25">A44</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:4_1_3_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:4_1_3"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:4_1_4"/><i><font color="Green" title="E8">A45</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:4_1_4"/><i><font color="Green" title="E9">A46</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:4_1_4"/><i><font color="Green" title="E10">A47</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c7">a''</font>, <font color="Maroon" title="c8">b''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E5:4_1_4"/><i><font color="Green" title="E11">A48</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a''</font>
 <b>and </b><br/><a NAME="E6:4_1_4"/><i><font color="Green" title="E12">A49</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b''</font>
 <b>and </b><br/><a NAME="E7:4_1_4"/><i><font color="Green" title="E13">A50</font></i>: 
<font color="Maroon" title="c7">a''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">b''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:4"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="xreal_1.html#E1:4_1_4"><i><font color="Green" title="E8">A45</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_4"><i><font color="Green" title="E9">A46</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">c'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E9:4_1_4"/><i><font color="Green" title="E14">A51</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font>
 <b>and </b><br/><a NAME="E10:4_1_4"/><i><font color="Green" title="E15">A52</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">c'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E11:4_1_4"/><i><font color="Green" title="E16">A53</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">c'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:4_1_4"><i><font color="Green" title="E8">A45</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_4"><i><font color="Green" title="E10">A47</font></i></a>, <a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c12">c''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E13:4_1_4"/><i><font color="Green" title="E17">A54</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b'</font>
 <b>and </b><br/><a NAME="E14:4_1_4"/><i><font color="Green" title="E18">A55</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">c''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E15:4_1_4"/><i><font color="Green" title="E19">A56</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:4_1_4"><i><font color="Green" title="E9">A46</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_4"><i><font color="Green" title="E10">A47</font></i></a>, <a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><a NAME="E16:4_1_4"/><i><font color="Green" title="E20">A57</font></i>: 
<font color="Maroon" title="c10">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E10:4_1_4"><i><font color="Green" title="E15">A52</font></i></a>, <a class="txt" href="xreal_1.html#E14:4_1_4"><i><font color="Green" title="E18">A55</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_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="E1:4_1_4_1_1"/>
( <font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">a'</font> or  not <font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">a'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_4_1_1_1"/><i><font color="Green" title="E21">A58</font></i>: 
<font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">a'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E22">A59</font></i>: 
<font color="Maroon" title="c9">a'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">c'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_4_1_1_1"/>
<font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">b'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:4_1_4"><i><font color="Green" title="E11">A48</font></i></a>, <a class="txt" href="xreal_1.html#E6:4_1_4"><i><font color="Green" title="E12">A49</font></i></a>, <a class="txt" href="xreal_1.html#E7:4_1_4"><i><font color="Green" title="E13">A50</font></i></a>, <a class="txt" href="xreal_1.html#E9:4_1_4"><i><font color="Green" title="E14">A51</font></i></a>, <a class="txt" href="xreal_1.html#E13:4_1_4"><i><font color="Green" title="E17">A54</font></i></a>, <a class="txt" href="xreal_1.html#E1:4_1_4_1_1_1"><i><font color="Green" title="E21">A58</font></i></a>, <a class="ref" href="arytm_1.html#T3" title="ARYTM_1:th.3">ARYTM_1:3</a></i>;<br/><a NAME="E4:4_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E23">A60</font></i>: 
<font color="Maroon" title="c11">b'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">c'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E5:4_1_4_1_1_1"/>
<font color="Maroon" title="c9">a'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">b'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c12">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:4_1_4"><i><font color="Green" title="E11">A48</font></i></a>, <a class="txt" href="xreal_1.html#E6:4_1_4"><i><font color="Green" title="E12">A49</font></i></a>, <a class="txt" href="xreal_1.html#E7:4_1_4"><i><font color="Green" title="E13">A50</font></i></a>, <a class="txt" href="xreal_1.html#E9:4_1_4"><i><font color="Green" title="E14">A51</font></i></a>, <a class="txt" href="xreal_1.html#E13:4_1_4"><i><font color="Green" title="E17">A54</font></i></a>, <a class="txt" href="xreal_1.html#E16:4_1_4"><i><font color="Green" title="E20">A57</font></i></a>, <a class="ref" href="arytm_1.html#T17" title="ARYTM_1:th.17">ARYTM_1:17</a></i>;<br/><b>hence </b><a NAME="E6:4_1_4_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="xreal_1.html#E11:4_1_4"><i><font color="Green" title="E16">A53</font></i></a>, <a class="txt" href="xreal_1.html#E15:4_1_4"><i><font color="Green" title="E19">A56</font></i></a>, <a class="txt" href="xreal_1.html#E16:4_1_4"><i><font color="Green" title="E20">A57</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_4_1_1_1"><i><font color="Green" title="E22">A59</font></i></a>, <a class="txt" href="xreal_1.html#E4:4_1_4_1_1_1"><i><font color="Green" title="E23">A60</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_4_1_1_2"/>
not <font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">a'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E21">A61</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">a'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E11:4_1_4"><i><font color="Green" title="E16">A53</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E22">A62</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_4_1_1_2_1_1"/>
( <font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">b'</font> or  not <font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">b'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">b'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_4_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">b'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">c'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_4_1_1_2_1_1_1"/><b>then </b>
( not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="xreal_1.html#E15:4_1_4"><i><font color="Green" title="E19">A56</font></i></a>, <a class="txt" href="xreal_1.html#E16:4_1_4"><i><font color="Green" title="E20">A57</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_4_1_1_2"><i><font color="Green" title="E22">A62</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E4:4_1_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_4_1_1_2_1_1_2"/>
not <font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">b'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E23">A63</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">b'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E15:4_1_4"><i><font color="Green" title="E19">A56</font></i></a>, <a class="txt" href="xreal_1.html#E16:4_1_4"><i><font color="Green" title="E20">A57</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E24">A64</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:4_1_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">b'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:4_1_4"><i><font color="Green" title="E11">A48</font></i></a>, <a class="txt" href="xreal_1.html#E6:4_1_4"><i><font color="Green" title="E12">A49</font></i></a>, <a class="txt" href="xreal_1.html#E7:4_1_4"><i><font color="Green" title="E13">A50</font></i></a>, <a class="txt" href="xreal_1.html#E9:4_1_4"><i><font color="Green" title="E14">A51</font></i></a>, <a class="txt" href="xreal_1.html#E13:4_1_4"><i><font color="Green" title="E17">A54</font></i></a>, <a class="ref" href="arytm_1.html#T16" title="ARYTM_1:th.16">ARYTM_1:16</a></i>;<br/><b>hence </b><a NAME="E5:4_1_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_4_1_1_2"><i><font color="Green" title="E21">A61</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_4_1_1_2"><i><font color="Green" title="E22">A62</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_4_1_1_2_1_1_2"><i><font color="Green" title="E23">A63</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_4_1_1_2_1_1_2"><i><font color="Green" title="E24">A64</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:4_1_4_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:4_1_4"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_5"><b>suppose </b></a><b>that </b><a NAME="E1:4_1_5"/><i><font color="Green" title="E8">A65</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E2:4_1_5"/><i><font color="Green" title="E9">A66</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:4_1_5"/><i><font color="Green" title="E10">A67</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><div class="add"><a NAME="E4:4_1_5"/>
( not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  )
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:4_1_5"><i><font color="Green" title="E8">A65</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_5"><i><font color="Green" title="E10">A67</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c8">c'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E6:4_1_5"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c7">a'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:4_1_5"/><i><font color="Green" title="E11">A68</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">c'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:4_1_5"/><i><font color="Green" title="E12">A69</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">c'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">b'</font>, <font color="Maroon" title="c10">c''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E10:4_1_5"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font>
 <b>and </b><br/><a NAME="E11:4_1_5"/><i><font color="Green" title="E13">A70</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">c''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E12:4_1_5"/><i><font color="Green" title="E14">A71</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:4_1_5"><i><font color="Green" title="E9">A66</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_5"><i><font color="Green" title="E10">A67</font></i></a>, <a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><a NAME="E13:4_1_5"/><i><font color="Green" title="E15">A72</font></i>: 
<font color="Maroon" title="c8">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E7:4_1_5"><i><font color="Green" title="E11">A68</font></i></a>, <a class="txt" href="xreal_1.html#E11:4_1_5"><i><font color="Green" title="E13">A70</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E14:4_1_5"/><i><font color="Green" title="E16">A73</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E8:4_1_5"><i><font color="Green" title="E12">A69</font></i></a>, <a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_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="E1:4_1_5_1_1"/>
( <font color="Maroon" title="c8">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">b'</font> or  not <font color="Maroon" title="c8">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">b'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_5_1_1_1"/>
<font color="Maroon" title="c8">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">b'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">b'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">c''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E13:4_1_5"><i><font color="Green" title="E15">A72</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_5_1_1_1"/><b>then </b>
( not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="xreal_1.html#E12:4_1_5"><i><font color="Green" title="E14">A71</font></i></a>, <a class="txt" href="xreal_1.html#E14:4_1_5"><i><font color="Green" title="E16">A73</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E4:4_1_5_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_5_1_1_2"/>
not <font color="Maroon" title="c8">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">b'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E17">A74</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c8">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">b'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E12:4_1_5"><i><font color="Green" title="E14">A71</font></i></a>, <a class="txt" href="xreal_1.html#E13:4_1_5"><i><font color="Green" title="E15">A72</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:4_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E18">A75</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:4_1_5_1_1_2"/><i><font color="Green" title="E19">A76</font></i>: 
<font color="Maroon" title="c8">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">b'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">c'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#T11" title="ARYTM_1:th.11">ARYTM_1:11</a></i>;<br/><a NAME="E5:4_1_5_1_1_2"/>
<font color="Maroon" title="c8">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">c'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">a'</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></i>;<br/><a NAME="E6:4_1_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">c'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">b'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c8">c'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">a'</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E4:4_1_5_1_1_2"><i><font color="Green" title="E19">A76</font></i></a>, <a class="ref" href="arytm_1.html#T3" title="ARYTM_1:th.3">ARYTM_1:3</a></i>;<br/><b>hence </b><a NAME="E7:4_1_5_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="xreal_1.html#E8:4_1_5"><i><font color="Green" title="E12">A69</font></i></a>, <a class="txt" href="xreal_1.html#E14:4_1_5"><i><font color="Green" title="E16">A73</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_5_1_1_2"><i><font color="Green" title="E17">A74</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_5_1_1_2"><i><font color="Green" title="E18">A75</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:4_1_5"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_6"><b>suppose </b></a><b>that </b><a NAME="E1:4_1_6"/><i><font color="Green" title="E8">A77</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E2:4_1_6"/><i><font color="Green" title="E9">A78</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E3:4_1_6"/><i><font color="Green" title="E10">A79</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c7">a''</font>, <font color="Maroon" title="c8">b''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E5:4_1_6"/><i><font color="Green" title="E11">A80</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c7">a''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:4_1_6"/><i><font color="Green" title="E12">A81</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">b''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:4_1_6"/><i><font color="Green" title="E13">A82</font></i>: 
<font color="Maroon" title="c8">b''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">a''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:4"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="xreal_1.html#E1:4_1_6"><i><font color="Green" title="E8">A77</font></i></a>, <a class="txt" href="xreal_1.html#E2:4_1_6"><i><font color="Green" title="E9">A78</font></i></a>, <a class="ref" href="xxreal_0.html#D5" title="XXREAL_0:def.5">XXREAL_0:def 5</a></i>;<br/><a NAME="E8:4_1_6"/>
( not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  )
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:4_1_6"><i><font color="Green" title="E8">A77</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_6"><i><font color="Green" title="E10">A79</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">c'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E10:4_1_6"/><i><font color="Green" title="E14">A83</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c9">a'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E11:4_1_6"/><i><font color="Green" title="E15">A84</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">c'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E12:4_1_6"/><i><font color="Green" title="E16">A85</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c9">a'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">c'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><a NAME="E13:4_1_6"/>
( not <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; not <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  )
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:4_1_6"><i><font color="Green" title="E9">A78</font></i></a>, <a class="txt" href="xreal_1.html#E3:4_1_6"><i><font color="Green" title="E10">A79</font></i></a>, <a class="ref" href="arytm_0.html#T5" title="ARYTM_0:th.5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c12">c''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E15:4_1_6"/><i><font color="Green" title="E17">A86</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">b'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E16:4_1_6"/><i><font color="Green" title="E18">A87</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">c''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E17:4_1_6"/><i><font color="Green" title="E19">A88</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c11">b'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">c''</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a></i>;<br/><a NAME="E18:4_1_6"/><i><font color="Green" title="E20">A89</font></i>: 
<font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:4_1_6"><i><font color="Green" title="E11">A80</font></i></a>, <a class="txt" href="xreal_1.html#E10:4_1_6"><i><font color="Green" title="E14">A83</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E19:4_1_6"/><i><font color="Green" title="E21">A90</font></i>: 
<font color="Maroon" title="c11">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E6:4_1_6"><i><font color="Green" title="E12">A81</font></i></a>, <a class="txt" href="xreal_1.html#E15:4_1_6"><i><font color="Green" title="E17">A86</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E20:4_1_6"/><i><font color="Green" title="E22">A91</font></i>: 
<font color="Maroon" title="c10">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E11:4_1_6"><i><font color="Green" title="E15">A84</font></i></a>, <a class="txt" href="xreal_1.html#E16:4_1_6"><i><font color="Green" title="E18">A87</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E21:4_1_6"/><b>then </b><i><font color="Green" title="E23">A92</font></i>: 
<font color="Maroon" title="c11">b'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">c'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">a'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">c''</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E7:4_1_6"><i><font color="Green" title="E13">A82</font></i></a>, <a class="txt" href="xreal_1.html#E18:4_1_6"><i><font color="Green" title="E20">A89</font></i></a>, <a class="txt" href="xreal_1.html#E19:4_1_6"><i><font color="Green" title="E21">A90</font></i></a>, <a class="ref" href="arytm_1.html#T7" title="ARYTM_1:th.7">ARYTM_1:7</a></i>;<br/><a NAME="E22:4_1_6"/><i><font color="Green" title="E24">A93</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E12:4_1_6"><i><font color="Green" title="E16">A85</font></i></a>, <a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E23:4_1_6"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E17:4_1_6"><i><font color="Green" title="E19">A88</font></i></a>, <a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E24:4_1_6"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="xreal_1.html#E12:4_1_6"><i><font color="Green" title="E16">A85</font></i></a>, <a class="txt" href="xreal_1.html#E17:4_1_6"><i><font color="Green" title="E19">A88</font></i></a>, <a class="txt" href="xreal_1.html#E20:4_1_6"><i><font color="Green" title="E22">A91</font></i></a>, <a class="txt" href="xreal_1.html#E21:4_1_6"><i><font color="Green" title="E23">A92</font></i></a>, <a class="txt" href="xreal_1.html#E22:4_1_6"><i><font color="Green" title="E24">A93</font></i></a>, <a class="txt" href="xreal_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
