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<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be    <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> ;<br/>

<a NAME="E1:28"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c4">x</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>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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">0</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="c5">y</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>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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">0</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="c6">z</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>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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">0</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="ref" href="numbers.html#D1" title="NUMBERS:def.1">NUMBERS:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( ( <font color="Maroon" title="c4">x</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="c5">y</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="c6">z</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="c4">x</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="c5">y</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="c6">z</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">0</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="c4">x</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="c5">y</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">0</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="c6">z</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="c4">x</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="c5">y</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">0</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="c6">z</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">0</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="c6">z</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="c5">y</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="c4">x</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">0</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="c4">x</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">0</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="c5">y</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="c6">z</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">0</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="c6">z</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="c5">y</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">0</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="c4">x</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">0</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="c4">x</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">0</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="c5">y</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">0</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="c6">z</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">0</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="arytm_0.html#E1:28"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c4">x</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="E2:28_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c5">y</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="E3:28_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c6">z</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">y'</font>, <font color="Maroon" title="c8">z'</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="E5:28_1_1"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font>
 <b>and </b><br/><a NAME="E6:28_1_1"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z'</font>
 <b>and </b><br/><a NAME="E7:28_1_1"/><i><font color="Green" title="E32">A7</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:28_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="arytm_0.html#E3:28_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">x'</font>, <font color="Maroon" title="c10">yz'</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="E9:28_1_1"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font>
 <b>and </b><br/><a NAME="E10:28_1_1"/><i><font color="Green" title="E34">A9</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">yz'</font>
 <b>and </b><br/><a NAME="E11:28_1_1"/><i><font color="Green" title="E35">A10</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">x''</font>, <font color="Maroon" title="c12">y''</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="E13:28_1_1"/><i><font color="Green" title="E36">A11</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x''</font>
 <b>and </b><br/><a NAME="E14:28_1_1"/><i><font color="Green" title="E37">A12</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y''</font>
 <b>and </b><br/><a NAME="E15:28_1_1"/><i><font color="Green" title="E38">A13</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">xy''</font>, <font color="Maroon" title="c14">z''</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="E17:28_1_1"/><i><font color="Green" title="E39">A14</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xy''</font>
 <b>and </b><br/><a NAME="E18:28_1_1"/><i><font color="Green" title="E40">A15</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z''</font>
 <b>and </b><br/><a NAME="E19:28_1_1"/><i><font color="Green" title="E41">A16</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xy''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_1"><i><font color="Green" title="E38">A13</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>thus </b><a NAME="E20:28_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_1"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="arytm_0.html#E14:28_1_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_1"><i><font color="Green" title="E38">A13</font></i></a>, <a class="txt" href="arytm_0.html#E17:28_1_1"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="arytm_0.html#E18:28_1_1"><i><font color="Green" title="E40">A15</font></i></a>, <a class="txt" href="arytm_0.html#E19:28_1_1"><i><font color="Green" title="E41">A16</font></i></a>, <a class="ref" href="arytm_2.html#T7" title="ARYTM_2:th.7">ARYTM_2:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_2"/><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon" title="c4">x</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="E2:28_1_2"/><i><font color="Green" title="E28">A18</font></i>: 
<font color="Maroon" title="c5">y</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="E3:28_1_2"/><i><font color="Green" title="E29">A19</font></i>: 
<font color="Maroon" title="c6">z</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">0</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">y'</font>, <font color="Maroon" title="c8">z'</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="E5:28_1_2"/><i><font color="Green" title="E30">A20</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font>
 <b>and </b><br/><a NAME="E6:28_1_2"/><i><font color="Green" title="E31">A21</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c8">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_2"/><i><font color="Green" title="E32">A22</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c8">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:28_1_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="arytm_0.html#E3:28_1_2"><i><font color="Green" title="E29">A19</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">x''</font>, <font color="Maroon" title="c10">y''</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="E9:28_1_2"/><i><font color="Green" title="E33">A23</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font>
 <b>and </b><br/><a NAME="E10:28_1_2"/><i><font color="Green" title="E34">A24</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y''</font>
 <b>and </b><br/><a NAME="E11:28_1_2"/><i><font color="Green" title="E35">A25</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">xy''</font>, <font color="Maroon" title="c12">z''</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="E13:28_1_2"/><i><font color="Green" title="E36">A26</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xy''</font>
 <b>and </b><br/><a NAME="E14:28_1_2"/><i><font color="Green" title="E37">A27</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c12">z''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E15:28_1_2"/><i><font color="Green" title="E38">A28</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xy''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_2"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_2"><i><font color="Green" title="E35">A25</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E16:28_1_2"/><i><font color="Green" title="E39">A29</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:28_1_2"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="arytm_0.html#E14:28_1_2"><i><font color="Green" title="E37">A27</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="28_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_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="E1:28_1_2_1_1"/>
( <font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font> or  not <font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_2_1_1_1"/><i><font color="Green" title="E40">A30</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font>
 ;<br/><div class="add"><a NAME="E2:28_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E41">A31</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_2"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">x'</font>, <font color="Maroon" title="c14">yz'</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="E4:28_1_2_1_1_1"/><i><font color="Green" title="E42">A32</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font>
 <b>and </b><br/><a NAME="E5:28_1_2_1_1_1"/><i><font color="Green" title="E43">A33</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">yz'</font>
 <b>and </b><br/><a NAME="E6:28_1_2_1_1_1"/><i><font color="Green" title="E44">A34</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>thus </b><a NAME="E7:28_1_2_1_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_2"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_2"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_2"><i><font color="Green" title="E34">A24</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_2"><i><font color="Green" title="E35">A25</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_2"><i><font color="Green" title="E36">A26</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_2"><i><font color="Green" title="E38">A28</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_2"><i><font color="Green" title="E39">A29</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_2_1_1_1"><i><font color="Green" title="E40">A30</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_2_1_1_1"><i><font color="Green" title="E41">A31</font></i></a>, <a class="txt" href="arytm_0.html#E4:28_1_2_1_1_1"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_2_1_1_1"><i><font color="Green" title="E43">A33</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_2_1_1_1"><i><font color="Green" title="E44">A34</font></i></a>, <a class="ref" href="arytm_1.html#T20" title="ARYTM_1:th.20">ARYTM_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_2_1_1_2"/><i><font color="Green" title="E40">A35</font></i>: 
not <font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font>
 ;<br/><div class="add"><a NAME="E2:28_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E41">A36</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c8">z'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_2"><i><font color="Green" title="E32">A22</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="E3:28_1_2_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">x'</font>, <font color="Maroon" title="c14">yz'</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="E5:28_1_2_1_1_2"/><i><font color="Green" title="E42">A37</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font>
 <b>and </b><br/><a NAME="E6:28_1_2_1_1_2"/><i><font color="Green" title="E43">A38</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c14">yz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_2_1_1_2"/><i><font color="Green" title="E44">A39</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c14">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c13">x'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">z'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E2:28_1_2_1_1_2"><i><font color="Green" title="E41">A36</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_2_1_1_2"><i><font color="Green" title="E43">A38</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_2_1_1_2"><i><font color="Green" title="E44">A39</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_2"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_2"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_2"><i><font color="Green" title="E34">A24</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_2"><i><font color="Green" title="E35">A25</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_2"><i><font color="Green" title="E36">A26</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_2"><i><font color="Green" title="E38">A28</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_2"><i><font color="Green" title="E39">A29</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_2_1_1_2"><i><font color="Green" title="E40">A35</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_2_1_1_2"><i><font color="Green" title="E42">A37</font></i></a>, <a class="ref" href="arytm_1.html#T21" title="ARYTM_1:th.21">ARYTM_1:21</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:28_1_2"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_3"/><i><font color="Green" title="E27">A40</font></i>: 
<font color="Maroon" title="c4">x</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="E2:28_1_3"/><i><font color="Green" title="E28">A41</font></i>: 
<font color="Maroon" title="c5">y</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">0</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="E3:28_1_3"/><i><font color="Green" title="E29">A42</font></i>: 
<font color="Maroon" title="c6">z</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">z'</font>, <font color="Maroon" title="c8">y'</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="E5:28_1_3"/><i><font color="Green" title="E30">A43</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z'</font>
 <b>and </b><br/><a NAME="E6:28_1_3"/><i><font color="Green" title="E31">A44</font></i>: 
<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c8">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_3"/><i><font color="Green" title="E32">A45</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c8">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:28_1_3"><i><font color="Green" title="E28">A41</font></i></a>, <a class="txt" href="arytm_0.html#E3:28_1_3"><i><font color="Green" title="E29">A42</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">x''</font>, <font color="Maroon" title="c10">y''</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="E9:28_1_3"/><i><font color="Green" title="E33">A46</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font>
 <b>and </b><br/><a NAME="E10:28_1_3"/><i><font color="Green" title="E34">A47</font></i>: 
<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c10">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E11:28_1_3"/><i><font color="Green" title="E35">A48</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_3"><i><font color="Green" title="E27">A40</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_3"><i><font color="Green" title="E28">A41</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E12:28_1_3"/><i><font color="Green" title="E36">A49</font></i>: 
<font color="Maroon" title="c8">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:28_1_3"><i><font color="Green" title="E31">A44</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_3"><i><font color="Green" title="E34">A47</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="28_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_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="E1:28_1_3_1_1"/>
( ( <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font> &amp; <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">z'</font> ) or ( <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font> &amp; not <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">z'</font> ) or ( not <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font> &amp; <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">z'</font> ) or ( not <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font> &amp; not <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">z'</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_3_1_1_1"/><i><font color="Green" title="E37">A50</font></i>: 
<font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font>
 <b>and </b><br/><a NAME="E2:28_1_3_1_1_1"/><i><font color="Green" title="E38">A51</font></i>: 
<font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">z'</font>
 ;<br/><div class="add"><a NAME="E3:28_1_3_1_1_1"/><i><font color="Green" title="E39">A52</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_3"><i><font color="Green" title="E32">A45</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_3_1_1_1"><i><font color="Green" title="E38">A51</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">x'</font>, <font color="Maroon" title="c12">yz'</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="E5:28_1_3_1_1_1"/><i><font color="Green" title="E40">A53</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x'</font>
 <b>and </b><br/><a NAME="E6:28_1_3_1_1_1"/><i><font color="Green" title="E41">A54</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yz'</font>
 <b>and </b><br/><a NAME="E7:28_1_3_1_1_1"/><i><font color="Green" title="E42">A55</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_3"><i><font color="Green" title="E27">A40</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E8:28_1_3_1_1_1"/><i><font color="Green" title="E43">A56</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:28_1_3"><i><font color="Green" title="E33">A46</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_3"><i><font color="Green" title="E35">A48</font></i></a>, <a class="txt" href="arytm_0.html#E12:28_1_3"><i><font color="Green" title="E36">A49</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_3_1_1_1"><i><font color="Green" title="E37">A50</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_3_1_1_1"><i><font color="Green" title="E40">A53</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">z''</font>, <font color="Maroon" title="c14">xy''</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="E10:28_1_3_1_1_1"/><i><font color="Green" title="E44">A57</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z''</font>
 <b>and </b><br/><a NAME="E11:28_1_3_1_1_1"/><i><font color="Green" title="E45">A58</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xy''</font>
 <b>and </b><br/><a NAME="E12:28_1_3_1_1_1"/><i><font color="Green" title="E46">A59</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">xy''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_3"><i><font color="Green" title="E29">A42</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>thus </b><a NAME="E13:28_1_3_1_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_3"><i><font color="Green" title="E30">A43</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_3"><i><font color="Green" title="E33">A46</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_3_1_1_1"><i><font color="Green" title="E37">A50</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_3_1_1_1"><i><font color="Green" title="E38">A51</font></i></a>, <a class="txt" href="arytm_0.html#E3:28_1_3_1_1_1"><i><font color="Green" title="E39">A52</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_3_1_1_1"><i><font color="Green" title="E40">A53</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_3_1_1_1"><i><font color="Green" title="E41">A54</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_3_1_1_1"><i><font color="Green" title="E42">A55</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_3_1_1_1"><i><font color="Green" title="E43">A56</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_3_1_1_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_3_1_1_1"><i><font color="Green" title="E45">A58</font></i></a>, <a class="txt" href="arytm_0.html#E12:28_1_3_1_1_1"><i><font color="Green" title="E46">A59</font></i></a>, <a class="ref" href="arytm_1.html#T12" title="ARYTM_1:th.12">ARYTM_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_3_1_1_2"/><i><font color="Green" title="E37">A60</font></i>: 
<font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font>
 <b>and </b><br/><a NAME="E2:28_1_3_1_1_2"/><i><font color="Green" title="E38">A61</font></i>: 
not <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">z'</font>
 ;<br/><div class="add"><a NAME="E3:28_1_3_1_1_2"/><i><font color="Green" title="E39">A62</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">z'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_3"><i><font color="Green" title="E32">A45</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_3_1_1_2"><i><font color="Green" title="E38">A61</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="E4:28_1_3_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">x'</font>, <font color="Maroon" title="c12">yz'</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="E6:28_1_3_1_1_2"/><i><font color="Green" title="E40">A63</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x'</font>
 <b>and </b><br/><a NAME="E7:28_1_3_1_1_2"/><i><font color="Green" title="E41">A64</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c12">yz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:28_1_3_1_1_2"/><i><font color="Green" title="E42">A65</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_3"><i><font color="Green" title="E27">A40</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E9:28_1_3_1_1_2"/><i><font color="Green" title="E43">A66</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:28_1_3"><i><font color="Green" title="E35">A48</font></i></a>, <a class="txt" href="arytm_0.html#E12:28_1_3"><i><font color="Green" title="E36">A49</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_3_1_1_2"><i><font color="Green" title="E37">A60</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">z''</font>, <font color="Maroon" title="c14">xy''</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="E11:28_1_3_1_1_2"/><i><font color="Green" title="E44">A67</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z''</font>
 <b>and </b><br/><a NAME="E12:28_1_3_1_1_2"/><i><font color="Green" title="E45">A68</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xy''</font>
 <b>and </b><br/><a NAME="E13:28_1_3_1_1_2"/><i><font color="Green" title="E46">A69</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">xy''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_3"><i><font color="Green" title="E29">A42</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c11">x'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">z'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_3_1_1_2"><i><font color="Green" title="E39">A62</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_3_1_1_2"><i><font color="Green" title="E41">A64</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_3_1_1_2"><i><font color="Green" title="E42">A65</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_3"><i><font color="Green" title="E30">A43</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_3"><i><font color="Green" title="E33">A46</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_3_1_1_2"><i><font color="Green" title="E37">A60</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_3_1_1_2"><i><font color="Green" title="E38">A61</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_3_1_1_2"><i><font color="Green" title="E40">A63</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_3_1_1_2"><i><font color="Green" title="E43">A66</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_3_1_1_2"><i><font color="Green" title="E44">A67</font></i></a>, <a class="txt" href="arytm_0.html#E12:28_1_3_1_1_2"><i><font color="Green" title="E45">A68</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_3_1_1_2"><i><font color="Green" title="E46">A69</font></i></a>, <a class="ref" href="arytm_1.html#T22" title="ARYTM_1:th.22">ARYTM_1:22</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_3_1_1_3"/><i><font color="Green" title="E37">A70</font></i>: 
not <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font>
 <b>and </b><br/><a NAME="E2:28_1_3_1_1_3"/><i><font color="Green" title="E38">A71</font></i>: 
<font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">z'</font>
 ;<br/><div class="add"><a NAME="E3:28_1_3_1_1_3"/><i><font color="Green" title="E39">A72</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x''</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:28_1_3"><i><font color="Green" title="E35">A48</font></i></a>, <a class="txt" href="arytm_0.html#E12:28_1_3"><i><font color="Green" title="E36">A49</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_3_1_1_3"><i><font color="Green" title="E37">A70</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="E4:28_1_3_1_1_3"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">z''</font>, <font color="Maroon" title="c12">yx''</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="E6:28_1_3_1_1_3"/><i><font color="Green" title="E40">A73</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z''</font>
 <b>and </b><br/><a NAME="E7:28_1_3_1_1_3"/><i><font color="Green" title="E41">A74</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c12">yx''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:28_1_3_1_1_3"/><i><font color="Green" title="E42">A75</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">yx''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_3"><i><font color="Green" title="E29">A42</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E9:28_1_3_1_1_3"/><i><font color="Green" title="E43">A76</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_3"><i><font color="Green" title="E32">A45</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_3_1_1_3"><i><font color="Green" title="E38">A71</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">x'</font>, <font color="Maroon" title="c14">zy''</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="E11:28_1_3_1_1_3"/><i><font color="Green" title="E44">A77</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font>
 <b>and </b><br/><a NAME="E12:28_1_3_1_1_3"/><i><font color="Green" title="E45">A78</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">zy''</font>
 <b>and </b><br/><a NAME="E13:28_1_3_1_1_3"/><i><font color="Green" title="E46">A79</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">zy''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_3"><i><font color="Green" title="E27">A40</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c11">z''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x''</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_3"><i><font color="Green" title="E30">A43</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_3"><i><font color="Green" title="E33">A46</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_3_1_1_3"><i><font color="Green" title="E37">A70</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_3_1_1_3"><i><font color="Green" title="E38">A71</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_3_1_1_3"><i><font color="Green" title="E40">A73</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_3_1_1_3"><i><font color="Green" title="E43">A76</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_3_1_1_3"><i><font color="Green" title="E44">A77</font></i></a>, <a class="txt" href="arytm_0.html#E12:28_1_3_1_1_3"><i><font color="Green" title="E45">A78</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_3_1_1_3"><i><font color="Green" title="E46">A79</font></i></a>, <a class="ref" href="arytm_1.html#T22" title="ARYTM_1:th.22">ARYTM_1:22</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_3_1_1_3"><i><font color="Green" title="E39">A72</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_3_1_1_3"><i><font color="Green" title="E41">A74</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_3_1_1_3"><i><font color="Green" title="E42">A75</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_3_1_1_4"/><i><font color="Green" title="E37">A80</font></i>: 
not <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font>
 <b>and </b><br/><a NAME="E2:28_1_3_1_1_4"/><i><font color="Green" title="E38">A81</font></i>: 
not <font color="Maroon" title="c8">y'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">z'</font>
 ;<br/><div class="add"><a NAME="E3:28_1_3_1_1_4"/><i><font color="Green" title="E39">A82</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x''</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:28_1_3"><i><font color="Green" title="E35">A48</font></i></a>, <a class="txt" href="arytm_0.html#E12:28_1_3"><i><font color="Green" title="E36">A49</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_3_1_1_4"><i><font color="Green" title="E37">A80</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="E4:28_1_3_1_1_4"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">z''</font>, <font color="Maroon" title="c12">yx''</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="E6:28_1_3_1_1_4"/><i><font color="Green" title="E40">A83</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z''</font>
 <b>and </b><br/><a NAME="E7:28_1_3_1_1_4"/><i><font color="Green" title="E41">A84</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c12">yx''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:28_1_3_1_1_4"/><i><font color="Green" title="E42">A85</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">yx''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_3"><i><font color="Green" title="E29">A42</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E9:28_1_3_1_1_4"/><i><font color="Green" title="E43">A86</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">z'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_3"><i><font color="Green" title="E32">A45</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_3_1_1_4"><i><font color="Green" title="E38">A81</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="E10:28_1_3_1_1_4"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">x'</font>, <font color="Maroon" title="c14">yz'</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="E12:28_1_3_1_1_4"/><i><font color="Green" title="E44">A87</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font>
 <b>and </b><br/><a NAME="E13:28_1_3_1_1_4"/><i><font color="Green" title="E45">A88</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c14">yz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E14:28_1_3_1_1_4"/><i><font color="Green" title="E46">A89</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c14">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_3"><i><font color="Green" title="E27">A40</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c13">x'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">z'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E9:28_1_3_1_1_4"><i><font color="Green" title="E43">A86</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_3_1_1_4"><i><font color="Green" title="E45">A88</font></i></a>, <a class="txt" href="arytm_0.html#E14:28_1_3_1_1_4"><i><font color="Green" title="E46">A89</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
<br/>.= 
<font color="Maroon" title="c11">z''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x''</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_3"><i><font color="Green" title="E30">A43</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_3"><i><font color="Green" title="E33">A46</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_3_1_1_4"><i><font color="Green" title="E37">A80</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_3_1_1_4"><i><font color="Green" title="E38">A81</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_3_1_1_4"><i><font color="Green" title="E40">A83</font></i></a>, <a class="txt" href="arytm_0.html#E12:28_1_3_1_1_4"><i><font color="Green" title="E44">A87</font></i></a>, <a class="ref" href="arytm_1.html#T23" title="ARYTM_1:th.23">ARYTM_1:23</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_3_1_1_4"><i><font color="Green" title="E39">A82</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_3_1_1_4"><i><font color="Green" title="E41">A84</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_3_1_1_4"><i><font color="Green" title="E42">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:28_1_3"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_4"/><i><font color="Green" title="E27">A90</font></i>: 
<font color="Maroon" title="c4">x</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="E2:28_1_4"/><i><font color="Green" title="E28">A91</font></i>: 
<font color="Maroon" title="c5">y</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">0</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="E3:28_1_4"/><i><font color="Green" title="E29">A92</font></i>: 
<font color="Maroon" title="c6">z</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">0</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="E4:28_1_4"/>
( (  not <font color="Maroon" title="c6">z</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  not <font color="Maroon" title="c5">y</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="c6">z</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  not <font color="Maroon" title="c5">y</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">0</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; (  not <font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E2:28_1_4"><i><font color="Green" title="E28">A91</font></i></a>, <a class="txt" href="arytm_0.html#E3:28_1_4"><i><font color="Green" title="E29">A92</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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">y'</font>, <font color="Maroon" title="c8">z'</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="E6:28_1_4"/><i><font color="Green" title="E30">A93</font></i>: 
<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c7">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_4"/><i><font color="Green" title="E31">A94</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c8">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:28_1_4"/><i><font color="Green" title="E32">A95</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">z'</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">x''</font>, <font color="Maroon" title="c10">y''</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="E10:28_1_4"/><i><font color="Green" title="E33">A96</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font>
 <b>and </b><br/><a NAME="E11:28_1_4"/><i><font color="Green" title="E34">A97</font></i>: 
<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c10">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E12:28_1_4"/><i><font color="Green" title="E35">A98</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_4"><i><font color="Green" title="E27">A90</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_4"><i><font color="Green" title="E28">A91</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E13:28_1_4"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E8:28_1_4"><i><font color="Green" title="E32">A95</font></i></a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">x'</font>, <font color="Maroon" title="c12">yz'</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="E15:28_1_4"/><i><font color="Green" title="E36">A99</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x'</font>
 <b>and </b><br/><a NAME="E16:28_1_4"/><i><font color="Green" title="E37">A100</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c12">yz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E17:28_1_4"/><i><font color="Green" title="E38">A101</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_4"><i><font color="Green" title="E27">A90</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E18:28_1_4"/><i><font color="Green" title="E39">A102</font></i>: 
<font color="Maroon" title="c7">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:28_1_4"><i><font color="Green" title="E30">A93</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_4"><i><font color="Green" title="E34">A97</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="28_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_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="E1:28_1_4_1_1"/>
( <font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font> or  not <font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_4_1_1_1"/><i><font color="Green" title="E40">A103</font></i>: 
<font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font>
 ;<br/><div class="add"><a NAME="E2:28_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E41">A104</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E12:28_1_4"><i><font color="Green" title="E35">A98</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">xy''</font>, <font color="Maroon" title="c14">z''</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="E4:28_1_4_1_1_1"/><i><font color="Green" title="E42">A105</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xy''</font>
 <b>and </b><br/><a NAME="E5:28_1_4_1_1_1"/><i><font color="Green" title="E43">A106</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c14">z''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:28_1_4_1_1_1"/><i><font color="Green" title="E44">A107</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xy''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c14">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_4"><i><font color="Green" title="E29">A92</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E7:28_1_4_1_1_1"/><i><font color="Green" title="E45">A108</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_4"><i><font color="Green" title="E31">A94</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_4_1_1_1"><i><font color="Green" title="E43">A106</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c11">x'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">z'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E8:28_1_4"><i><font color="Green" title="E32">A95</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_4"><i><font color="Green" title="E37">A100</font></i></a>, <a class="txt" href="arytm_0.html#E17:28_1_4"><i><font color="Green" title="E38">A101</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E10:28_1_4"><i><font color="Green" title="E33">A96</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_4"><i><font color="Green" title="E36">A99</font></i></a>, <a class="txt" href="arytm_0.html#E18:28_1_4"><i><font color="Green" title="E39">A102</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_4_1_1_1"><i><font color="Green" title="E40">A103</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_4_1_1_1"><i><font color="Green" title="E41">A104</font></i></a>, <a class="txt" href="arytm_0.html#E4:28_1_4_1_1_1"><i><font color="Green" title="E42">A105</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_4_1_1_1"><i><font color="Green" title="E44">A107</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_4_1_1_1"><i><font color="Green" title="E45">A108</font></i></a>, <a class="ref" href="arytm_1.html#T24" title="ARYTM_1:th.24">ARYTM_1:24</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_4_1_1_2"/><i><font color="Green" title="E40">A109</font></i>: 
not <font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x''</font>
 ;<br/><div class="add"><a NAME="E2:28_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E41">A110</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c10">y''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x''</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E12:28_1_4"><i><font color="Green" title="E35">A98</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="E3:28_1_4_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:28_1_4_1_1_2"/><b>then </b>
( (  not <font color="Maroon" title="c6">z</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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="c6">z</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">0</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; (  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E3:28_1_4"><i><font color="Green" title="E29">A92</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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="c13">xy''</font>, <font color="Maroon" title="c14">z''</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="E6:28_1_4_1_1_2"/><i><font color="Green" title="E42">A111</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c13">xy''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_4_1_1_2"/><i><font color="Green" title="E43">A112</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c14">z''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:28_1_4_1_1_2"/><i><font color="Green" title="E44">A113</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c13">xy''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">z''</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E9:28_1_4_1_1_2"/><i><font color="Green" title="E45">A114</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_4"><i><font color="Green" title="E31">A94</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_4_1_1_2"><i><font color="Green" title="E43">A112</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E10:28_1_4_1_1_2"/><i><font color="Green" title="E46">A115</font></i>: 
<font color="Maroon" title="c12">yz'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E8:28_1_4"><i><font color="Green" title="E32">A95</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_4"><i><font color="Green" title="E37">A100</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E11:28_1_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c12">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E18:28_1_4"><i><font color="Green" title="E39">A102</font></i></a>, <a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></i>;<br/><a NAME="E12:28_1_4_1_1_2"/><b>then </b>
not <font color="Maroon" title="c12">yz'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">x'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E10:28_1_4"><i><font color="Green" title="E33">A96</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_4"><i><font color="Green" title="E36">A99</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_4_1_1_2"><i><font color="Green" title="E40">A109</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 href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">x'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E17:28_1_4"><i><font color="Green" title="E38">A101</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_4_1_1_2"><i><font color="Green" title="E46">A115</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c14">z''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">y''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x''</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E10:28_1_4"><i><font color="Green" title="E33">A96</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_4"><i><font color="Green" title="E36">A99</font></i></a>, <a class="txt" href="arytm_0.html#E18:28_1_4"><i><font color="Green" title="E39">A102</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_4_1_1_2"><i><font color="Green" title="E40">A109</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_4_1_1_2"><i><font color="Green" title="E45">A114</font></i></a>, <a class="ref" href="arytm_1.html#T13" title="ARYTM_1:th.13">ARYTM_1:13</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E2:28_1_4_1_1_2"><i><font color="Green" title="E41">A110</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_4_1_1_2"><i><font color="Green" title="E42">A111</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_4_1_1_2"><i><font color="Green" title="E44">A113</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E20:28_1_4"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_5"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_5"/><i><font color="Green" title="E27">A116</font></i>: 
<font color="Maroon" title="c6">z</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="E2:28_1_5"/><i><font color="Green" title="E28">A117</font></i>: 
<font color="Maroon" title="c5">y</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="E3:28_1_5"/><i><font color="Green" title="E29">A118</font></i>: 
<font color="Maroon" title="c4">x</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">0</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">y'</font>, <font color="Maroon" title="c8">x'</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="E5:28_1_5"/><i><font color="Green" title="E30">A119</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font>
 <b>and </b><br/><a NAME="E6:28_1_5"/><i><font color="Green" title="E31">A120</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c8">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_5"/><i><font color="Green" title="E32">A121</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c8">x'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:28_1_5"><i><font color="Green" title="E28">A117</font></i></a>, <a class="txt" href="arytm_0.html#E3:28_1_5"><i><font color="Green" title="E29">A118</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">z''</font>, <font color="Maroon" title="c10">y''</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="E9:28_1_5"/><i><font color="Green" title="E33">A122</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z''</font>
 <b>and </b><br/><a NAME="E10:28_1_5"/><i><font color="Green" title="E34">A123</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y''</font>
 <b>and </b><br/><a NAME="E11:28_1_5"/><i><font color="Green" title="E35">A124</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_5"><i><font color="Green" title="E27">A116</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_5"><i><font color="Green" title="E28">A117</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">zy''</font>, <font color="Maroon" title="c12">x''</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="E13:28_1_5"/><i><font color="Green" title="E36">A125</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">zy''</font>
 <b>and </b><br/><a NAME="E14:28_1_5"/><i><font color="Green" title="E37">A126</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c12">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E15:28_1_5"/><i><font color="Green" title="E38">A127</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">zy''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_5"><i><font color="Green" title="E29">A118</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_5"><i><font color="Green" title="E35">A124</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E16:28_1_5"/><i><font color="Green" title="E39">A128</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:28_1_5"><i><font color="Green" title="E31">A120</font></i></a>, <a class="txt" href="arytm_0.html#E14:28_1_5"><i><font color="Green" title="E37">A126</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="28_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_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="E1:28_1_5_1_1"/>
( <font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font> or  not <font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_5_1_1_1"/><i><font color="Green" title="E40">A129</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font>
 ;<br/><div class="add"><a NAME="E2:28_1_5_1_1_1"/><b>then </b><i><font color="Green" title="E41">A130</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">x'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_5"><i><font color="Green" title="E32">A121</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">z'</font>, <font color="Maroon" title="c14">yx'</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="E4:28_1_5_1_1_1"/><i><font color="Green" title="E42">A131</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z'</font>
 <b>and </b><br/><a NAME="E5:28_1_5_1_1_1"/><i><font color="Green" title="E43">A132</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">yx'</font>
 <b>and </b><br/><a NAME="E6:28_1_5_1_1_1"/><i><font color="Green" title="E44">A133</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">yx'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_5"><i><font color="Green" title="E27">A116</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>thus </b><a NAME="E7:28_1_5_1_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_5"><i><font color="Green" title="E30">A119</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_5"><i><font color="Green" title="E33">A122</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_5"><i><font color="Green" title="E34">A123</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_5"><i><font color="Green" title="E35">A124</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_5"><i><font color="Green" title="E36">A125</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_5"><i><font color="Green" title="E38">A127</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_5"><i><font color="Green" title="E39">A128</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_5_1_1_1"><i><font color="Green" title="E40">A129</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_5_1_1_1"><i><font color="Green" title="E41">A130</font></i></a>, <a class="txt" href="arytm_0.html#E4:28_1_5_1_1_1"><i><font color="Green" title="E42">A131</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_5_1_1_1"><i><font color="Green" title="E43">A132</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_5_1_1_1"><i><font color="Green" title="E44">A133</font></i></a>, <a class="ref" href="arytm_1.html#T20" title="ARYTM_1:th.20">ARYTM_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_5_1_1_2"/><i><font color="Green" title="E40">A134</font></i>: 
not <font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font>
 ;<br/><div class="add"><a NAME="E2:28_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E41">A135</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_5"><i><font color="Green" title="E32">A121</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="E3:28_1_5_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">z'</font>, <font color="Maroon" title="c14">yx'</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="E5:28_1_5_1_1_2"/><i><font color="Green" title="E42">A136</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z'</font>
 <b>and </b><br/><a NAME="E6:28_1_5_1_1_2"/><i><font color="Green" title="E43">A137</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c14">yx'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_5_1_1_2"/><i><font color="Green" title="E44">A138</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c14">yx'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_5"><i><font color="Green" title="E27">A116</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> = 
<font color="Maroon" title="c13">z'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E2:28_1_5_1_1_2"><i><font color="Green" title="E41">A135</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_5_1_1_2"><i><font color="Green" title="E43">A137</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_5_1_1_2"><i><font color="Green" title="E44">A138</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_5"><i><font color="Green" title="E30">A119</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_5"><i><font color="Green" title="E33">A122</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_5"><i><font color="Green" title="E34">A123</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_5"><i><font color="Green" title="E35">A124</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_5"><i><font color="Green" title="E36">A125</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_5"><i><font color="Green" title="E38">A127</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_5"><i><font color="Green" title="E39">A128</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_5_1_1_2"><i><font color="Green" title="E40">A134</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_5_1_1_2"><i><font color="Green" title="E42">A136</font></i></a>, <a class="ref" href="arytm_1.html#T21" title="ARYTM_1:th.21">ARYTM_1:21</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:28_1_5"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_6"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_6"/><i><font color="Green" title="E27">A139</font></i>: 
<font color="Maroon" title="c4">x</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">0</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="E2:28_1_6"/><i><font color="Green" title="E28">A140</font></i>: 
<font color="Maroon" title="c5">y</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="E3:28_1_6"/><i><font color="Green" title="E29">A141</font></i>: 
<font color="Maroon" title="c6">z</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">0</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">y'</font>, <font color="Maroon" title="c8">z'</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="E5:28_1_6"/><i><font color="Green" title="E30">A142</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font>
 <b>and </b><br/><a NAME="E6:28_1_6"/><i><font color="Green" title="E31">A143</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c8">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_6"/><i><font color="Green" title="E32">A144</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c8">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:28_1_6"><i><font color="Green" title="E28">A140</font></i></a>, <a class="txt" href="arytm_0.html#E3:28_1_6"><i><font color="Green" title="E29">A141</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">x''</font>, <font color="Maroon" title="c10">y''</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="E9:28_1_6"/><i><font color="Green" title="E33">A145</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c9">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E10:28_1_6"/><i><font color="Green" title="E34">A146</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y''</font>
 <b>and </b><br/><a NAME="E11:28_1_6"/><i><font color="Green" title="E35">A147</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c9">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_6"><i><font color="Green" title="E27">A139</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_6"><i><font color="Green" title="E28">A140</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_6_1_1"/>
( ( <font color="Maroon" title="c9">x''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font> &amp; <font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font> ) or ( not <font color="Maroon" title="c9">x''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font> &amp; <font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font> ) or ( not <font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font> &amp; <font color="Maroon" title="c9">x''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font> ) or ( not <font color="Maroon" title="c9">x''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font> &amp; not <font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_6_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_6_1_1_1"/><i><font color="Green" title="E36">A148</font></i>: 
<font color="Maroon" title="c9">x''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font>
 <b>and </b><br/><a NAME="E2:28_1_6_1_1_1"/><i><font color="Green" title="E37">A149</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font>
 ;<br/><div class="add"><a NAME="E3:28_1_6_1_1_1"/><i><font color="Green" title="E38">A150</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_6"><i><font color="Green" title="E32">A144</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_6_1_1_1"><i><font color="Green" title="E37">A149</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">x'</font>, <font color="Maroon" title="c12">yz'</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="E5:28_1_6_1_1_1"/><i><font color="Green" title="E39">A151</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c11">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:28_1_6_1_1_1"/><i><font color="Green" title="E40">A152</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yz'</font>
 <b>and </b><br/><a NAME="E7:28_1_6_1_1_1"/><i><font color="Green" title="E41">A153</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yz'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c11">x'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_6"><i><font color="Green" title="E27">A139</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E8:28_1_6_1_1_1"/><i><font color="Green" title="E42">A154</font></i>: 
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:28_1_6"><i><font color="Green" title="E33">A145</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_6_1_1_1"><i><font color="Green" title="E39">A151</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:28_1_6_1_1_1"/><b>then </b><i><font color="Green" title="E43">A155</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">x'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_6"><i><font color="Green" title="E30">A142</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_6"><i><font color="Green" title="E34">A146</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_6"><i><font color="Green" title="E35">A147</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_6_1_1_1"><i><font color="Green" title="E36">A148</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">z''</font>, <font color="Maroon" title="c14">xy''</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="E11:28_1_6_1_1_1"/><i><font color="Green" title="E44">A156</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c13">z''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E12:28_1_6_1_1_1"/><i><font color="Green" title="E45">A157</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xy''</font>
 <b>and </b><br/><a NAME="E13:28_1_6_1_1_1"/><i><font color="Green" title="E46">A158</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xy''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c13">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_6"><i><font color="Green" title="E29">A141</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E14:28_1_6_1_1_1"/>
<font color="Maroon" title="c8">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:28_1_6"><i><font color="Green" title="E31">A143</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_6_1_1_1"><i><font color="Green" title="E44">A156</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E15:28_1_6_1_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_6"><i><font color="Green" title="E30">A142</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_6"><i><font color="Green" title="E34">A146</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_6_1_1_1"><i><font color="Green" title="E36">A148</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_6_1_1_1"><i><font color="Green" title="E37">A149</font></i></a>, <a class="txt" href="arytm_0.html#E3:28_1_6_1_1_1"><i><font color="Green" title="E38">A150</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_6_1_1_1"><i><font color="Green" title="E40">A152</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_6_1_1_1"><i><font color="Green" title="E41">A153</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_6_1_1_1"><i><font color="Green" title="E42">A154</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_6_1_1_1"><i><font color="Green" title="E43">A155</font></i></a>, <a class="txt" href="arytm_0.html#E12:28_1_6_1_1_1"><i><font color="Green" title="E45">A157</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_6_1_1_1"><i><font color="Green" title="E46">A158</font></i></a>, <a class="ref" href="arytm_1.html#T25" title="ARYTM_1:th.25">ARYTM_1:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_6_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_6_1_1_2"/><i><font color="Green" title="E36">A159</font></i>: 
not <font color="Maroon" title="c9">x''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font>
 <b>and </b><br/><a NAME="E2:28_1_6_1_1_2"/><i><font color="Green" title="E37">A160</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font>
 ;<br/><div class="add"><a NAME="E3:28_1_6_1_1_2"/><i><font color="Green" title="E38">A161</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_6"><i><font color="Green" title="E32">A144</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_6_1_1_2"><i><font color="Green" title="E37">A160</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">x'</font>, <font color="Maroon" title="c12">yz'</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="E5:28_1_6_1_1_2"/><i><font color="Green" title="E39">A162</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c11">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:28_1_6_1_1_2"/><i><font color="Green" title="E40">A163</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yz'</font>
 <b>and </b><br/><a NAME="E7:28_1_6_1_1_2"/><i><font color="Green" title="E41">A164</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yz'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c11">x'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_6"><i><font color="Green" title="E27">A139</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E8:28_1_6_1_1_2"/><i><font color="Green" title="E42">A165</font></i>: 
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:28_1_6"><i><font color="Green" title="E33">A145</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_6_1_1_2"><i><font color="Green" title="E39">A162</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:28_1_6_1_1_2"/><i><font color="Green" title="E43">A166</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y''</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:28_1_6"><i><font color="Green" title="E35">A147</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_6_1_1_2"><i><font color="Green" title="E36">A159</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="E10:28_1_6_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:28_1_6_1_1_2"/><b>then </b>
( (  not <font color="Maroon" title="c6">z</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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="c6">z</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">0</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; (  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E3:28_1_6"><i><font color="Green" title="E29">A141</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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="c13">z''</font>, <font color="Maroon" title="c14">yx'</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="E13:28_1_6_1_1_2"/><i><font color="Green" title="E44">A167</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c13">z''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E14:28_1_6_1_1_2"/><i><font color="Green" title="E45">A168</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c14">yx'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E15:28_1_6_1_1_2"/><i><font color="Green" title="E46">A169</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c13">z''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">yx'</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E16:28_1_6_1_1_2"/><i><font color="Green" title="E47">A170</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:28_1_6"><i><font color="Green" title="E31">A143</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_6_1_1_2"><i><font color="Green" title="E44">A167</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E17:28_1_6_1_1_2"/>
<font color="Maroon" title="c12">yz'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_6_1_1_2"><i><font color="Green" title="E38">A161</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_6_1_1_2"><i><font color="Green" title="E40">A163</font></i></a>, <a class="ref" href="arytm_1.html#T11" title="ARYTM_1:th.11">ARYTM_1:11</a></i>;<br/><a NAME="E18:28_1_6_1_1_2"/><b>then </b>
not <font color="Maroon" title="c11">x'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c12">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_6"><i><font color="Green" title="E30">A142</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_6"><i><font color="Green" title="E34">A146</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_6_1_1_2"><i><font color="Green" title="E36">A159</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_6_1_1_2"><i><font color="Green" title="E42">A165</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 href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c11">x'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">z'</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_6_1_1_2"><i><font color="Green" title="E38">A161</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_6_1_1_2"><i><font color="Green" title="E40">A163</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_6_1_1_2"><i><font color="Green" title="E41">A164</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">x''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y''</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c13">z''</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_6"><i><font color="Green" title="E30">A142</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_6"><i><font color="Green" title="E34">A146</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_6_1_1_2"><i><font color="Green" title="E36">A159</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_6_1_1_2"><i><font color="Green" title="E37">A160</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_6_1_1_2"><i><font color="Green" title="E42">A165</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_6_1_1_2"><i><font color="Green" title="E47">A170</font></i></a>, <a class="ref" href="arytm_1.html#T14" title="ARYTM_1:th.14">ARYTM_1:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E9:28_1_6_1_1_2"><i><font color="Green" title="E43">A166</font></i></a>, <a class="txt" href="arytm_0.html#E14:28_1_6_1_1_2"><i><font color="Green" title="E45">A168</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_6_1_1_2"><i><font color="Green" title="E46">A169</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_6_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_6_1_1_3"/><i><font color="Green" title="E36">A171</font></i>: 
not <font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font>
 <b>and </b><br/><a NAME="E2:28_1_6_1_1_3"/><i><font color="Green" title="E37">A172</font></i>: 
<font color="Maroon" title="c9">x''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font>
 ;<br/><div class="add"><a NAME="E3:28_1_6_1_1_3"/><i><font color="Green" title="E38">A173</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:28_1_6"><i><font color="Green" title="E35">A147</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_6_1_1_3"><i><font color="Green" title="E37">A172</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">z''</font>, <font color="Maroon" title="c12">yx''</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="E5:28_1_6_1_1_3"/><i><font color="Green" title="E39">A174</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c11">z''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:28_1_6_1_1_3"/><i><font color="Green" title="E40">A175</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yx''</font>
 <b>and </b><br/><a NAME="E7:28_1_6_1_1_3"/><i><font color="Green" title="E41">A176</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yx''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c11">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_6"><i><font color="Green" title="E29">A141</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E8:28_1_6_1_1_3"/><i><font color="Green" title="E42">A177</font></i>: 
<font color="Maroon" title="c11">z''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:28_1_6"><i><font color="Green" title="E31">A143</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_6_1_1_3"><i><font color="Green" title="E39">A174</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:28_1_6_1_1_3"/><i><font color="Green" title="E43">A178</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c8">z'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_6"><i><font color="Green" title="E32">A144</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_6_1_1_3"><i><font color="Green" title="E36">A171</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="E10:28_1_6_1_1_3"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:28_1_6_1_1_3"/><b>then </b>
( (  not <font color="Maroon" title="c4">x</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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="c4">x</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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">0</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; (  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E1:28_1_6"><i><font color="Green" title="E27">A139</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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="c13">x'</font>, <font color="Maroon" title="c14">yz''</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="E13:28_1_6_1_1_3"/><i><font color="Green" title="E44">A179</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c13">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E14:28_1_6_1_1_3"/><i><font color="Green" title="E45">A180</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c14">yz''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E15:28_1_6_1_1_3"/><i><font color="Green" title="E46">A181</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c13">x'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">yz''</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E16:28_1_6_1_1_3"/><i><font color="Green" title="E47">A182</font></i>: 
<font color="Maroon" title="c9">x''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:28_1_6"><i><font color="Green" title="E33">A145</font></i></a>, <a class="txt" href="arytm_0.html#E13:28_1_6_1_1_3"><i><font color="Green" title="E44">A179</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E17:28_1_6_1_1_3"/>
<font color="Maroon" title="c12">yx''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_6_1_1_3"><i><font color="Green" title="E38">A173</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_6_1_1_3"><i><font color="Green" title="E40">A175</font></i></a>, <a class="ref" href="arytm_1.html#T11" title="ARYTM_1:th.11">ARYTM_1:11</a></i>;<br/><a NAME="E18:28_1_6_1_1_3"/><b>then </b><i><font color="Green" title="E48">A183</font></i>: 
not <font color="Maroon" title="c11">z''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c12">yx''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_6"><i><font color="Green" title="E30">A142</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_6"><i><font color="Green" title="E34">A146</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_6_1_1_3"><i><font color="Green" title="E36">A171</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_6_1_1_3"><i><font color="Green" title="E42">A177</font></i></a>, <a class="ref" href="arytm_1.html#T3" title="ARYTM_1:th.3">ARYTM_1:3</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c13">x'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E9:28_1_6_1_1_3"><i><font color="Green" title="E43">A178</font></i></a>, <a class="txt" href="arytm_0.html#E14:28_1_6_1_1_3"><i><font color="Green" title="E45">A180</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_6_1_1_3"><i><font color="Green" title="E46">A181</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c11">z''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">y''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x''</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_6"><i><font color="Green" title="E30">A142</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_6"><i><font color="Green" title="E34">A146</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_6_1_1_3"><i><font color="Green" title="E36">A171</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_6_1_1_3"><i><font color="Green" title="E37">A172</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_6_1_1_3"><i><font color="Green" title="E42">A177</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_6_1_1_3"><i><font color="Green" title="E47">A182</font></i></a>, <a class="ref" href="arytm_1.html#T14" title="ARYTM_1:th.14">ARYTM_1:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_6_1_1_3"><i><font color="Green" title="E38">A173</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_6_1_1_3"><i><font color="Green" title="E40">A175</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_6_1_1_3"><i><font color="Green" title="E41">A176</font></i></a>, <a class="txt" href="arytm_0.html#E18:28_1_6_1_1_3"><i><font color="Green" title="E48">A183</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_6_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_6_1_1_4"/><i><font color="Green" title="E36">A184</font></i>: 
not <font color="Maroon" title="c9">x''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font>
 <b>and </b><br/><a NAME="E2:28_1_6_1_1_4"/><i><font color="Green" title="E37">A185</font></i>: 
not <font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y'</font>
 ;<br/><div class="add"><a NAME="E3:28_1_6_1_1_4"/><i><font color="Green" title="E38">A186</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c8">z'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_6"><i><font color="Green" title="E32">A144</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_6_1_1_4"><i><font color="Green" title="E37">A185</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="E4:28_1_6_1_1_4"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:28_1_6_1_1_4"/><b>then </b>
( (  not <font color="Maroon" title="c4">x</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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="c4">x</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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">0</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; (  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E1:28_1_6"><i><font color="Green" title="E27">A139</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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">x'</font>, <font color="Maroon" title="c12">yz''</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="E7:28_1_6_1_1_4"/><i><font color="Green" title="E39">A187</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c11">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:28_1_6_1_1_4"/><i><font color="Green" title="E40">A188</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c12">yz''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E9:28_1_6_1_1_4"/><i><font color="Green" title="E41">A189</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c11">x'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">yz''</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E10:28_1_6_1_1_4"/><i><font color="Green" title="E42">A190</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y''</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:28_1_6"><i><font color="Green" title="E35">A147</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_6_1_1_4"><i><font color="Green" title="E36">A184</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="E11:28_1_6_1_1_4"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:28_1_6_1_1_4"/><b>then </b>
( (  not <font color="Maroon" title="c6">z</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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="c6">z</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">0</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; (  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E3:28_1_6"><i><font color="Green" title="E29">A141</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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="c13">z''</font>, <font color="Maroon" title="c14">yx'</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="E14:28_1_6_1_1_4"/><i><font color="Green" title="E43">A191</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c13">z''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E15:28_1_6_1_1_4"/><i><font color="Green" title="E44">A192</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c14">yx'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E16:28_1_6_1_1_4"/><i><font color="Green" title="E45">A193</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c13">z''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">yx'</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E17:28_1_6_1_1_4"/><i><font color="Green" title="E46">A194</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:28_1_6"><i><font color="Green" title="E31">A143</font></i></a>, <a class="txt" href="arytm_0.html#E14:28_1_6_1_1_4"><i><font color="Green" title="E43">A191</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E18:28_1_6_1_1_4"/><i><font color="Green" title="E47">A195</font></i>: 
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:28_1_6"><i><font color="Green" title="E33">A145</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_6_1_1_4"><i><font color="Green" title="E39">A187</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z'</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c11">x'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_6_1_1_4"><i><font color="Green" title="E38">A186</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_6_1_1_4"><i><font color="Green" title="E40">A188</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_6_1_1_4"><i><font color="Green" title="E41">A189</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">x''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y''</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c13">z''</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E5:28_1_6"><i><font color="Green" title="E30">A142</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_6"><i><font color="Green" title="E34">A146</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_6_1_1_4"><i><font color="Green" title="E36">A184</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_6_1_1_4"><i><font color="Green" title="E37">A185</font></i></a>, <a class="txt" href="arytm_0.html#E17:28_1_6_1_1_4"><i><font color="Green" title="E46">A194</font></i></a>, <a class="txt" href="arytm_0.html#E18:28_1_6_1_1_4"><i><font color="Green" title="E47">A195</font></i></a>, <a class="ref" href="arytm_1.html#T15" title="ARYTM_1:th.15">ARYTM_1:15</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E10:28_1_6_1_1_4"><i><font color="Green" title="E42">A190</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_6_1_1_4"><i><font color="Green" title="E44">A192</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_6_1_1_4"><i><font color="Green" title="E45">A193</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:28_1_6"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_7"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_7"/><i><font color="Green" title="E27">A196</font></i>: 
<font color="Maroon" title="c6">z</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="E2:28_1_7"/><i><font color="Green" title="E28">A197</font></i>: 
<font color="Maroon" title="c5">y</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">0</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="E3:28_1_7"/><i><font color="Green" title="E29">A198</font></i>: 
<font color="Maroon" title="c4">x</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">0</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="E4:28_1_7"/>
( (  not <font color="Maroon" title="c4">x</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  not <font color="Maroon" title="c5">y</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="c4">x</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  not <font color="Maroon" title="c5">y</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">0</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; (  not <font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E2:28_1_7"><i><font color="Green" title="E28">A197</font></i></a>, <a class="txt" href="arytm_0.html#E3:28_1_7"><i><font color="Green" title="E29">A198</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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">y'</font>, <font color="Maroon" title="c8">x'</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="E6:28_1_7"/><i><font color="Green" title="E30">A199</font></i>: 
<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c7">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_7"/><i><font color="Green" title="E31">A200</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c8">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:28_1_7"/><i><font color="Green" title="E32">A201</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">x'</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">z''</font>, <font color="Maroon" title="c10">y''</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="E10:28_1_7"/><i><font color="Green" title="E33">A202</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z''</font>
 <b>and </b><br/><a NAME="E11:28_1_7"/><i><font color="Green" title="E34">A203</font></i>: 
<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c10">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E12:28_1_7"/><i><font color="Green" title="E35">A204</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_7"><i><font color="Green" title="E27">A196</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_7"><i><font color="Green" title="E28">A197</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E13:28_1_7"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E8:28_1_7"><i><font color="Green" title="E32">A201</font></i></a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">z'</font>, <font color="Maroon" title="c12">yx'</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="E15:28_1_7"/><i><font color="Green" title="E36">A205</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z'</font>
 <b>and </b><br/><a NAME="E16:28_1_7"/><i><font color="Green" title="E37">A206</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c12">yx'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E17:28_1_7"/><i><font color="Green" title="E38">A207</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">yx'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:28_1_7"><i><font color="Green" title="E27">A196</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E18:28_1_7"/><i><font color="Green" title="E39">A208</font></i>: 
<font color="Maroon" title="c7">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:28_1_7"><i><font color="Green" title="E30">A199</font></i></a>, <a class="txt" href="arytm_0.html#E11:28_1_7"><i><font color="Green" title="E34">A203</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="28_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_7_1_1"/>
( <font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z''</font> or  not <font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z''</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_7_1_1_1"/><i><font color="Green" title="E40">A209</font></i>: 
<font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z''</font>
 ;<br/><div class="add"><a NAME="E2:28_1_7_1_1_1"/><b>then </b><i><font color="Green" title="E41">A210</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E12:28_1_7"><i><font color="Green" title="E35">A204</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">zy''</font>, <font color="Maroon" title="c14">x''</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="E4:28_1_7_1_1_1"/><i><font color="Green" title="E42">A211</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">zy''</font>
 <b>and </b><br/><a NAME="E5:28_1_7_1_1_1"/><i><font color="Green" title="E43">A212</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c14">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:28_1_7_1_1_1"/><i><font color="Green" title="E44">A213</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">zy''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c14">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:28_1_7"><i><font color="Green" title="E29">A198</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E7:28_1_7_1_1_1"/><i><font color="Green" title="E45">A214</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_7"><i><font color="Green" title="E31">A200</font></i></a>, <a class="txt" href="arytm_0.html#E5:28_1_7_1_1_1"><i><font color="Green" title="E43">A212</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> = 
<font color="Maroon" title="c11">z'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">x'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E8:28_1_7"><i><font color="Green" title="E32">A201</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_7"><i><font color="Green" title="E37">A206</font></i></a>, <a class="txt" href="arytm_0.html#E17:28_1_7"><i><font color="Green" title="E38">A207</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E10:28_1_7"><i><font color="Green" title="E33">A202</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_7"><i><font color="Green" title="E36">A205</font></i></a>, <a class="txt" href="arytm_0.html#E18:28_1_7"><i><font color="Green" title="E39">A208</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_7_1_1_1"><i><font color="Green" title="E40">A209</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_7_1_1_1"><i><font color="Green" title="E41">A210</font></i></a>, <a class="txt" href="arytm_0.html#E4:28_1_7_1_1_1"><i><font color="Green" title="E42">A211</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_7_1_1_1"><i><font color="Green" title="E44">A213</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_7_1_1_1"><i><font color="Green" title="E45">A214</font></i></a>, <a class="ref" href="arytm_1.html#T24" title="ARYTM_1:th.24">ARYTM_1:24</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_7_1_1_2"/><i><font color="Green" title="E40">A215</font></i>: 
not <font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z''</font>
 ;<br/><div class="add"><a NAME="E2:28_1_7_1_1_2"/><b>then </b><i><font color="Green" title="E41">A216</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c10">y''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">z''</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E12:28_1_7"><i><font color="Green" title="E35">A204</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="E3:28_1_7_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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">0</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="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:28_1_7_1_1_2"/><b>then </b>
( (  not <font color="Maroon" title="c4">x</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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="c4">x</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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">0</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; (  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E3:28_1_7"><i><font color="Green" title="E29">A198</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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="c13">zy''</font>, <font color="Maroon" title="c14">x''</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="E6:28_1_7_1_1_2"/><i><font color="Green" title="E42">A217</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c13">zy''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_7_1_1_2"/><i><font color="Green" title="E43">A218</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c14">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:28_1_7_1_1_2"/><i><font color="Green" title="E44">A219</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c13">zy''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">x''</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E9:28_1_7_1_1_2"/><i><font color="Green" title="E45">A220</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_7"><i><font color="Green" title="E31">A200</font></i></a>, <a class="txt" href="arytm_0.html#E7:28_1_7_1_1_2"><i><font color="Green" title="E43">A218</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E10:28_1_7_1_1_2"/><i><font color="Green" title="E46">A221</font></i>: 
<font color="Maroon" title="c12">yx'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E8:28_1_7"><i><font color="Green" title="E32">A201</font></i></a>, <a class="txt" href="arytm_0.html#E16:28_1_7"><i><font color="Green" title="E37">A206</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E11:28_1_7_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c12">yx'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E18:28_1_7"><i><font color="Green" title="E39">A208</font></i></a>, <a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></i>;<br/><a NAME="E12:28_1_7_1_1_2"/><b>then </b>
not <font color="Maroon" title="c12">yx'</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E10:28_1_7"><i><font color="Green" title="E33">A202</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_7"><i><font color="Green" title="E36">A205</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_7_1_1_2"><i><font color="Green" title="E40">A215</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 href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">z'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E17:28_1_7"><i><font color="Green" title="E38">A207</font></i></a>, <a class="txt" href="arytm_0.html#E10:28_1_7_1_1_2"><i><font color="Green" title="E46">A221</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c14">x''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">y''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">z''</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E10:28_1_7"><i><font color="Green" title="E33">A202</font></i></a>, <a class="txt" href="arytm_0.html#E15:28_1_7"><i><font color="Green" title="E36">A205</font></i></a>, <a class="txt" href="arytm_0.html#E18:28_1_7"><i><font color="Green" title="E39">A208</font></i></a>, <a class="txt" href="arytm_0.html#E1:28_1_7_1_1_2"><i><font color="Green" title="E40">A215</font></i></a>, <a class="txt" href="arytm_0.html#E9:28_1_7_1_1_2"><i><font color="Green" title="E45">A220</font></i></a>, <a class="ref" href="arytm_1.html#T13" title="ARYTM_1:th.13">ARYTM_1:13</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E2:28_1_7_1_1_2"><i><font color="Green" title="E41">A216</font></i></a>, <a class="txt" href="arytm_0.html#E6:28_1_7_1_1_2"><i><font color="Green" title="E42">A217</font></i></a>, <a class="txt" href="arytm_0.html#E8:28_1_7_1_1_2"><i><font color="Green" title="E44">A219</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E20:28_1_7"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_8"><b>suppose </b></a><b>that </b><a NAME="E1:28_1_8"/><i><font color="Green" title="E27">A222</font></i>: 
<font color="Maroon" title="c4">x</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">0</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="E2:28_1_8"/><i><font color="Green" title="E28">A223</font></i>: 
<font color="Maroon" title="c5">y</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">0</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="E3:28_1_8"/><i><font color="Green" title="E29">A224</font></i>: 
<font color="Maroon" title="c6">z</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">0</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="E4:28_1_8"/>
( (  not <font color="Maroon" title="c6">z</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  not <font color="Maroon" title="c5">y</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="c6">z</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  not <font color="Maroon" title="c5">y</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">0</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; (  not <font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E2:28_1_8"><i><font color="Green" title="E28">A223</font></i></a>, <a class="txt" href="arytm_0.html#E3:28_1_8"><i><font color="Green" title="E29">A224</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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">y'</font>, <font color="Maroon" title="c8">z'</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="E6:28_1_8"/><i><font color="Green" title="E30">A225</font></i>: 
<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c7">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:28_1_8"/><i><font color="Green" title="E31">A226</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c8">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:28_1_8"/><i><font color="Green" title="E32">A227</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">z'</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E9:28_1_8"/>
( (  not <font color="Maroon" title="c4">x</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  not <font color="Maroon" title="c5">y</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="c4">x</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  not <font color="Maroon" title="c5">y</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">0</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; (  not <font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E1:28_1_8"><i><font color="Green" title="E27">A222</font></i></a>, <a class="txt" href="arytm_0.html#E2:28_1_8"><i><font color="Green" title="E28">A223</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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">x''</font>, <font color="Maroon" title="c10">y''</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="E11:28_1_8"/><i><font color="Green" title="E33">A228</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c9">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E12:28_1_8"/><i><font color="Green" title="E34">A229</font></i>: 
<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c10">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E13:28_1_8"/><i><font color="Green" title="E35">A230</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">y''</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E14:28_1_8"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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">0</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="arytm_0.html#E8:28_1_8"><i><font color="Green" title="E32">A227</font></i></a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E15:28_1_8"/><b>then </b>
( (  not <font color="Maroon" title="c4">x</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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="c4">x</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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">0</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; (  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c4">x</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">0</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="arytm_0.html#E1:28_1_8"><i><font color="Green" title="E27">A222</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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">x'</font>, <font color="Maroon" title="c12">yz'</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="E17:28_1_8"/><i><font color="Green" title="E36">A231</font></i>: 
<font color="Maroon" title="c4">x</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">0,<font color="Maroon" title="c11">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E18:28_1_8"/><i><font color="Green" title="E37">A232</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c12">yz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E19:28_1_8"/><i><font color="Green" title="E38">A233</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c11">x'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">yz'</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E20:28_1_8"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">0</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="arytm_0.html#E13:28_1_8"><i><font color="Green" title="E35">A230</font></i></a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E21:28_1_8"/><b>then </b>
( (  not <font color="Maroon" title="c6">z</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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="c6">z</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  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">0</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; (  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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  not <font color="Maroon" title="c6">z</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">0</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="arytm_0.html#E3:28_1_8"><i><font color="Green" title="E29">A224</font></i></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</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="c13">xy''</font>, <font color="Maroon" title="c14">z''</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="E23:28_1_8"/><i><font color="Green" title="E39">A234</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">0,<font color="Maroon" title="c13">xy''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E24:28_1_8"/><i><font color="Green" title="E40">A235</font></i>: 
<font color="Maroon" title="c6">z</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">0,<font color="Maroon" title="c14">z''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E25:28_1_8"/><i><font color="Green" title="E41">A236</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</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">0,<span class="p2">(<span class="default"><font color="Maroon" title="c13">xy''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">z''</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" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E26:28_1_8"/><i><font color="Green" title="E42">A237</font></i>: 
<font color="Maroon" title="c8">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:28_1_8"><i><font color="Green" title="E31">A226</font></i></a>, <a class="txt" href="arytm_0.html#E24:28_1_8"><i><font color="Green" title="E40">A235</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E27:28_1_8"/><i><font color="Green" title="E43">A238</font></i>: 
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:28_1_8"><i><font color="Green" title="E33">A228</font></i></a>, <a class="txt" href="arytm_0.html#E17:28_1_8"><i><font color="Green" title="E36">A231</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E28:28_1_8"/><i><font color="Green" title="E44">A239</font></i>: 
<font color="Maroon" title="c7">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:28_1_8"><i><font color="Green" title="E30">A225</font></i></a>, <a class="txt" href="arytm_0.html#E12:28_1_8"><i><font color="Green" title="E34">A229</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c11">x'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E8:28_1_8"><i><font color="Green" title="E32">A227</font></i></a>, <a class="txt" href="arytm_0.html#E18:28_1_8"><i><font color="Green" title="E37">A232</font></i></a>, <a class="txt" href="arytm_0.html#E19:28_1_8"><i><font color="Green" title="E38">A233</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c14">z''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">y''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c9">x''</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_0.html#E26:28_1_8"><i><font color="Green" title="E42">A237</font></i></a>, <a class="txt" href="arytm_0.html#E27:28_1_8"><i><font color="Green" title="E43">A238</font></i></a>, <a class="txt" href="arytm_0.html#E28:28_1_8"><i><font color="Green" title="E44">A239</font></i></a>, <a class="ref" href="arytm_2.html#T7" title="ARYTM_2:th.7">ARYTM_2:7</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E13:28_1_8"><i><font color="Green" title="E35">A230</font></i></a>, <a class="txt" href="arytm_0.html#E23:28_1_8"><i><font color="Green" title="E39">A234</font></i></a>, <a class="txt" href="arytm_0.html#E25:28_1_8"><i><font color="Green" title="E41">A236</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
