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<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or ( <font color="Maroon" title="c3">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="c4">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="c5">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="c3">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="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c4">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="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c3">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="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c5">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="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c3">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="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">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="c5">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="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">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="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">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="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( (  not <font color="Maroon" title="c3">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="c4">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="c5">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; (  not <font color="Maroon" title="c3">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="c4">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="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c3">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="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c5">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; (  not <font color="Maroon" title="c3">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="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c4">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="c5">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; (  not <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c4">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="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c5">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; (  not <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>suppose </b></a><a NAME="E1:19_1_1"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
0
<b>by </b><i><a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th13</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:19_1_2"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c3">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="E2:19_1_2"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">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="E3:19_1_2"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c5">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="c6">x'</font>, <font color="Maroon" title="c7">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="E5:19_1_2"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 <b>and </b><br/><a NAME="E6:19_1_2"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c4">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="E7:19_1_2"/><i><font color="Green" title="E26">A7</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_2"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">x''</font>, <font color="Maroon" title="c9">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="E9:19_1_2"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x''</font>
 <b>and </b><br/><a NAME="E10:19_1_2"/><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c5">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="E11:19_1_2"/><i><font color="Green" title="E29">A10</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_2"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">xy'</font>, <font color="Maroon" title="c11">xz'</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="E13:19_1_2"/><i><font color="Green" title="E30">A11</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">xy'</font>
 <b>and </b><br/><a NAME="E14:19_1_2"/><i><font color="Green" title="E31">A12</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xz'</font>
 <b>and </b><br/><a NAME="E15:19_1_2"/><i><font color="Green" title="E32">A13</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">xy'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c11">xz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:19_1_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="arytm_0.html#E11:19_1_2"><i><font color="Green" title="E29">A10</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="c12">y''</font>, <font color="Maroon" title="c13">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="E17:19_1_2"/><i><font color="Green" title="E33">A14</font></i>: 
<font color="Maroon" title="c4">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="E18:19_1_2"/><i><font color="Green" title="E34">A15</font></i>: 
<font color="Maroon" title="c5">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="E19:19_1_2"/><i><font color="Green" title="E35">A16</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c13">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:19_1_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_2"><i><font color="Green" title="E23">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="c14">x'''</font>, <font color="Maroon" title="c15">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="E21:19_1_2"/><i><font color="Green" title="E36">A17</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'''</font>
 <b>and </b><br/><a NAME="E22:19_1_2"/><i><font color="Green" title="E37">A18</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yz'</font>
 <b>and </b><br/><a NAME="E23:19_1_2"/><i><font color="Green" title="E38">A19</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c15">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_2"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="arytm_0.html#E19:19_1_2"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>thus </b><a NAME="E24:19_1_2"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:19_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_2"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="arytm_0.html#E9:19_1_2"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="arytm_0.html#E10:19_1_2"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="arytm_0.html#E11:19_1_2"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="arytm_0.html#E13:19_1_2"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_2"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="arytm_0.html#E15:19_1_2"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="arytm_0.html#E17:19_1_2"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="arytm_0.html#E18:19_1_2"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="arytm_0.html#E19:19_1_2"><i><font color="Green" title="E35">A16</font></i></a>, <a class="txt" href="arytm_0.html#E21:19_1_2"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="arytm_0.html#E22:19_1_2"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="arytm_0.html#E23:19_1_2"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="arytm_2.html#T14" title="ARYTM_2:th.14">ARYTM_2:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:19_1_3"/><i><font color="Green" title="E21">A20</font></i>: 
( <font color="Maroon" title="c3">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="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 <b>and </b><br/><a NAME="E2:19_1_3"/><i><font color="Green" title="E22">A21</font></i>: 
<font color="Maroon" title="c4">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="E3:19_1_3"/><i><font color="Green" title="E23">A22</font></i>: 
<font color="Maroon" title="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_3"/><i><font color="Green" title="E24">A23</font></i>: 
<font color="Maroon" title="c5">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#E3:19_1_3"><i><font color="Green" title="E23">A22</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">x'</font>, <font color="Maroon" title="c7">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="E6:19_1_3"/><i><font color="Green" title="E25">A24</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 <b>and </b><br/><a NAME="E7:19_1_3"/><i><font color="Green" title="E26">A25</font></i>: 
<font color="Maroon" title="c4">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="E8:19_1_3"/><i><font color="Green" title="E27">A26</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_3"><i><font color="Green" title="E21">A20</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_3"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">x''</font>, <font color="Maroon" title="c9">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="E10:19_1_3"/><i><font color="Green" title="E28">A27</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x''</font>
 <b>and </b><br/><a NAME="E11:19_1_3"/><i><font color="Green" title="E29">A28</font></i>: 
<font color="Maroon" title="c5">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="c9">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E12:19_1_3"/><i><font color="Green" title="E30">A29</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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">x''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</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#E1:19_1_3"><i><font color="Green" title="E21">A20</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_3"><i><font color="Green" title="E24">A23</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">y''</font>, <font color="Maroon" title="c11">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="E14:19_1_3"/><i><font color="Green" title="E31">A30</font></i>: 
<font color="Maroon" title="c4">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="E15:19_1_3"/><i><font color="Green" title="E32">A31</font></i>: 
<font color="Maroon" title="c5">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="E16:19_1_3"/><i><font color="Green" title="E33">A32</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</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="c11">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:19_1_3"><i><font color="Green" title="E22">A21</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_3"><i><font color="Green" title="E24">A23</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E17:19_1_3"/><i><font color="Green" title="E34">A33</font></i>: 
<font color="Maroon" title="c9">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:19_1_3"><i><font color="Green" title="E29">A28</font></i></a>, <a class="txt" href="arytm_0.html#E15:19_1_3"><i><font color="Green" title="E32">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E18:19_1_3"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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#E12:19_1_3"><i><font color="Green" title="E30">A29</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="c12">xy'</font>, <font color="Maroon" title="c13">xz'</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="E20:19_1_3"/><i><font color="Green" title="E35">A34</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">xy'</font>
 <b>and </b><br/><a NAME="E21:19_1_3"/><i><font color="Green" title="E36">A35</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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">xz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E22:19_1_3"/><i><font color="Green" title="E37">A36</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">xy'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c13">xz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_3"><i><font color="Green" title="E27">A26</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="19_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_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="E1:19_1_3_1_1"/>
( <font color="Maroon" title="c11">z''</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="c11">z''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_3_1_1_1"/><i><font color="Green" title="E38">A37</font></i>: 
<font color="Maroon" title="c11">z''</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="E2:19_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E39">A38</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</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="c11">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E16:19_1_3"><i><font color="Green" title="E33">A32</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="c14">x'''</font>, <font color="Maroon" title="c15">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="E4:19_1_3_1_1_1"/><i><font color="Green" title="E40">A39</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'''</font>
 <b>and </b><br/><a NAME="E5:19_1_3_1_1_1"/><i><font color="Green" title="E41">A40</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yz'</font>
 <b>and </b><br/><a NAME="E6:19_1_3_1_1_1"/><i><font color="Green" title="E42">A41</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c15">yz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_3"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">y'</font></span>)</span> <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_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">z'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E6:19_1_3"><i><font color="Green" title="E25">A24</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_3"><i><font color="Green" title="E26">A25</font></i></a>, <a class="txt" href="arytm_0.html#E10:19_1_3"><i><font color="Green" title="E28">A27</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_3"><i><font color="Green" title="E31">A30</font></i></a>, <a class="txt" href="arytm_0.html#E17:19_1_3"><i><font color="Green" title="E34">A33</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_3_1_1_1"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_3_1_1_1"><i><font color="Green" title="E39">A38</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_3_1_1_1"><i><font color="Green" title="E40">A39</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_3_1_1_1"><i><font color="Green" title="E41">A40</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_3_1_1_1"><i><font color="Green" title="E42">A41</font></i></a>, <a class="ref" href="arytm_1.html#T26" title="ARYTM_1:th.26">ARYTM_1:26</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_3"><i><font color="Green" title="E27">A26</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_3"><i><font color="Green" title="E30">A29</font></i></a>, <a class="txt" href="arytm_0.html#E20:19_1_3"><i><font color="Green" title="E35">A34</font></i></a>, <a class="txt" href="arytm_0.html#E21:19_1_3"><i><font color="Green" title="E36">A35</font></i></a>, <a class="txt" href="arytm_0.html#E22:19_1_3"><i><font color="Green" title="E37">A36</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="19_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_3_1_1_2"/><i><font color="Green" title="E38">A42</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="c10">y''</font>
 ;<br/><div class="add"><a NAME="E2:19_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E39">A43</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c11">z''</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#E16:19_1_3"><i><font color="Green" title="E33">A32</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:19_1_3_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c14">x'''</font>, <font color="Maroon" title="c15">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="E5:19_1_3_1_1_2"/><i><font color="Green" title="E40">A44</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'''</font>
 <b>and </b><br/><a NAME="E6:19_1_3_1_1_2"/><i><font color="Green" title="E41">A45</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c15">yz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:19_1_3_1_1_2"/><i><font color="Green" title="E42">A46</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c15">yz'</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#E1:19_1_3"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">z''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y''</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#E2:19_1_3_1_1_2"><i><font color="Green" title="E39">A43</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_3_1_1_2"><i><font color="Green" title="E41">A45</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_3_1_1_2"><i><font color="Green" title="E42">A46</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">(<span class="default"><font color="Maroon" title="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">y'</font></span>)</span> <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_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">z'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_3"><i><font color="Green" title="E21">A20</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_3"><i><font color="Green" title="E25">A24</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_3"><i><font color="Green" title="E26">A25</font></i></a>, <a class="txt" href="arytm_0.html#E10:19_1_3"><i><font color="Green" title="E28">A27</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_3"><i><font color="Green" title="E31">A30</font></i></a>, <a class="txt" href="arytm_0.html#E17:19_1_3"><i><font color="Green" title="E34">A33</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_3_1_1_2"><i><font color="Green" title="E38">A42</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_3_1_1_2"><i><font color="Green" title="E40">A44</font></i></a>, <a class="ref" href="arytm_1.html#T27" title="ARYTM_1:th.27">ARYTM_1:27</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_3"><i><font color="Green" title="E27">A26</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_3"><i><font color="Green" title="E30">A29</font></i></a>, <a class="txt" href="arytm_0.html#E20:19_1_3"><i><font color="Green" title="E35">A34</font></i></a>, <a class="txt" href="arytm_0.html#E21:19_1_3"><i><font color="Green" title="E36">A35</font></i></a>, <a class="txt" href="arytm_0.html#E22:19_1_3"><i><font color="Green" title="E37">A36</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="E24:19_1_3"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:19_1_4"/><i><font color="Green" title="E21">A47</font></i>: 
( <font color="Maroon" title="c3">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="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 <b>and </b><br/><a NAME="E2:19_1_4"/><i><font color="Green" title="E22">A48</font></i>: 
<font color="Maroon" title="c5">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="E3:19_1_4"/><i><font color="Green" title="E23">A49</font></i>: 
<font color="Maroon" title="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_4"/><i><font color="Green" title="E24">A50</font></i>: 
<font color="Maroon" title="c4">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#E3:19_1_4"><i><font color="Green" title="E23">A49</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">x'</font>, <font color="Maroon" title="c7">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="E6:19_1_4"/><i><font color="Green" title="E25">A51</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 <b>and </b><br/><a NAME="E7:19_1_4"/><i><font color="Green" title="E26">A52</font></i>: 
<font color="Maroon" title="c5">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="E8:19_1_4"/><i><font color="Green" title="E27">A53</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_4"><i><font color="Green" title="E21">A47</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_4"><i><font color="Green" title="E22">A48</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">x''</font>, <font color="Maroon" title="c9">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="E10:19_1_4"/><i><font color="Green" title="E28">A54</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x''</font>
 <b>and </b><br/><a NAME="E11:19_1_4"/><i><font color="Green" title="E29">A55</font></i>: 
<font color="Maroon" title="c4">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="c9">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E12:19_1_4"/><i><font color="Green" title="E30">A56</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c8">x''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">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#E1:19_1_4"><i><font color="Green" title="E21">A47</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_4"><i><font color="Green" title="E24">A50</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">z''</font>, <font color="Maroon" title="c11">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="E14:19_1_4"/><i><font color="Green" title="E31">A57</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z''</font>
 <b>and </b><br/><a NAME="E15:19_1_4"/><i><font color="Green" title="E32">A58</font></i>: 
<font color="Maroon" title="c4">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="c11">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E16:19_1_4"/><i><font color="Green" title="E33">A59</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c11">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:19_1_4"><i><font color="Green" title="E22">A48</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_4"><i><font color="Green" title="E24">A50</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E17:19_1_4"/><i><font color="Green" title="E34">A60</font></i>: 
<font color="Maroon" title="c9">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:19_1_4"><i><font color="Green" title="E29">A55</font></i></a>, <a class="txt" href="arytm_0.html#E15:19_1_4"><i><font color="Green" title="E32">A58</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E18:19_1_4"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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#E12:19_1_4"><i><font color="Green" title="E30">A56</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="c12">xz'</font>, <font color="Maroon" title="c13">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="E20:19_1_4"/><i><font color="Green" title="E35">A61</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">xz'</font>
 <b>and </b><br/><a NAME="E21:19_1_4"/><i><font color="Green" title="E36">A62</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="E22:19_1_4"/><i><font color="Green" title="E37">A63</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">xz'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c13">xy'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_4"><i><font color="Green" title="E27">A53</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="19_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_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="E1:19_1_4_1_1"/>
( <font color="Maroon" title="c11">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">z''</font> or  not <font color="Maroon" title="c11">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">z''</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_4_1_1_1"/><i><font color="Green" title="E38">A64</font></i>: 
<font color="Maroon" title="c11">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">z''</font>
 ;<br/><div class="add"><a NAME="E2:19_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E39">A65</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E16:19_1_4"><i><font color="Green" title="E33">A59</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="c14">x'''</font>, <font color="Maroon" title="c15">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="E4:19_1_4_1_1_1"/><i><font color="Green" title="E40">A66</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'''</font>
 <b>and </b><br/><a NAME="E5:19_1_4_1_1_1"/><i><font color="Green" title="E41">A67</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">zy'</font>
 <b>and </b><br/><a NAME="E6:19_1_4_1_1_1"/><i><font color="Green" title="E42">A68</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">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">z</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c15">zy'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_4"><i><font color="Green" title="E21">A47</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">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">z</font>,<font color="Maroon" title="c4">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">z'</font></span>)</span> <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_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E6:19_1_4"><i><font color="Green" title="E25">A51</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_4"><i><font color="Green" title="E26">A52</font></i></a>, <a class="txt" href="arytm_0.html#E10:19_1_4"><i><font color="Green" title="E28">A54</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_4"><i><font color="Green" title="E31">A57</font></i></a>, <a class="txt" href="arytm_0.html#E17:19_1_4"><i><font color="Green" title="E34">A60</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_4_1_1_1"><i><font color="Green" title="E38">A64</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_4_1_1_1"><i><font color="Green" title="E39">A65</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_4_1_1_1"><i><font color="Green" title="E40">A66</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_4_1_1_1"><i><font color="Green" title="E41">A67</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_4_1_1_1"><i><font color="Green" title="E42">A68</font></i></a>, <a class="ref" href="arytm_1.html#T26" title="ARYTM_1:th.26">ARYTM_1:26</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_4"><i><font color="Green" title="E27">A53</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_4"><i><font color="Green" title="E30">A56</font></i></a>, <a class="txt" href="arytm_0.html#E20:19_1_4"><i><font color="Green" title="E35">A61</font></i></a>, <a class="txt" href="arytm_0.html#E21:19_1_4"><i><font color="Green" title="E36">A62</font></i></a>, <a class="txt" href="arytm_0.html#E22:19_1_4"><i><font color="Green" title="E37">A63</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="19_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_4_1_1_2"/><i><font color="Green" title="E38">A69</font></i>: 
not <font color="Maroon" title="c11">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">z''</font>
 ;<br/><div class="add"><a NAME="E2:19_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E39">A70</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">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="c11">y''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">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#E16:19_1_4"><i><font color="Green" title="E33">A59</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="E3:19_1_4_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">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/><b>then consider </b><font color="Maroon" title="c14">x'''</font>, <font color="Maroon" title="c15">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="E5:19_1_4_1_1_2"/><i><font color="Green" title="E40">A71</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'''</font>
 <b>and </b><br/><a NAME="E6:19_1_4_1_1_2"/><i><font color="Green" title="E41">A72</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">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="c15">zy'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:19_1_4_1_1_2"/><i><font color="Green" title="E42">A73</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">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">z</font>,<font color="Maroon" title="c4">y</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="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c15">zy'</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#E1:19_1_4"><i><font color="Green" title="E21">A47</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">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">z</font>,<font color="Maroon" title="c4">y</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="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">y''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">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#E2:19_1_4_1_1_2"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_4_1_1_2"><i><font color="Green" title="E41">A72</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_4_1_1_2"><i><font color="Green" title="E42">A73</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">(<span class="default"><font color="Maroon" title="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">z'</font></span>)</span> <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_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_4"><i><font color="Green" title="E21">A47</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_4"><i><font color="Green" title="E25">A51</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_4"><i><font color="Green" title="E26">A52</font></i></a>, <a class="txt" href="arytm_0.html#E10:19_1_4"><i><font color="Green" title="E28">A54</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_4"><i><font color="Green" title="E31">A57</font></i></a>, <a class="txt" href="arytm_0.html#E17:19_1_4"><i><font color="Green" title="E34">A60</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_4_1_1_2"><i><font color="Green" title="E38">A69</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_4_1_1_2"><i><font color="Green" title="E40">A71</font></i></a>, <a class="ref" href="arytm_1.html#T27" title="ARYTM_1:th.27">ARYTM_1:27</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_4"><i><font color="Green" title="E27">A53</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_4"><i><font color="Green" title="E30">A56</font></i></a>, <a class="txt" href="arytm_0.html#E20:19_1_4"><i><font color="Green" title="E35">A61</font></i></a>, <a class="txt" href="arytm_0.html#E21:19_1_4"><i><font color="Green" title="E36">A62</font></i></a>, <a class="txt" href="arytm_0.html#E22:19_1_4"><i><font color="Green" title="E37">A63</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="E24:19_1_4"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_5"><b>suppose </b></a><b>that </b><a NAME="E1:19_1_5"/><i><font color="Green" title="E21">A74</font></i>: 
( <font color="Maroon" title="c3">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="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 <b>and </b><br/><a NAME="E2:19_1_5"/><i><font color="Green" title="E22">A75</font></i>: 
<font color="Maroon" title="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E3:19_1_5"/><i><font color="Green" title="E23">A76</font></i>: 
<font color="Maroon" title="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_5"/><i><font color="Green" title="E24">A77</font></i>: 
<font color="Maroon" title="c4">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#E2:19_1_5"><i><font color="Green" title="E22">A75</font></i></a></i>;<br/><a NAME="E5:19_1_5"/><i><font color="Green" title="E25">A78</font></i>: 
<font color="Maroon" title="c5">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#E3:19_1_5"><i><font color="Green" title="E23">A76</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">x'</font>, <font color="Maroon" title="c7">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="E7:19_1_5"/><i><font color="Green" title="E26">A79</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 <b>and </b><br/><a NAME="E8:19_1_5"/><i><font color="Green" title="E27">A80</font></i>: 
<font color="Maroon" title="c4">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="E9:19_1_5"/><i><font color="Green" title="E28">A81</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</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#E1:19_1_5"><i><font color="Green" title="E21">A74</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_5"><i><font color="Green" title="E24">A77</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">x''</font>, <font color="Maroon" title="c9">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="E11:19_1_5"/><i><font color="Green" title="E29">A82</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x''</font>
 <b>and </b><br/><a NAME="E12:19_1_5"/><i><font color="Green" title="E30">A83</font></i>: 
<font color="Maroon" title="c5">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="c9">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E13:19_1_5"/><i><font color="Green" title="E31">A84</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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">x''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</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#E1:19_1_5"><i><font color="Green" title="E21">A74</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_5"><i><font color="Green" title="E25">A78</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><a NAME="E14:19_1_5"/>
(  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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;  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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#E9:19_1_5"><i><font color="Green" title="E28">A81</font></i></a>, <a class="txt" href="arytm_0.html#E13:19_1_5"><i><font color="Green" title="E31">A84</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="E15:19_1_5"/><b>then </b>
( (  not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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; (  not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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> ) &amp; (  not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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> or  not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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>by </b><i><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="c10">xy'</font>, <font color="Maroon" title="c11">xz'</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="E17:19_1_5"/><i><font color="Green" title="E32">A85</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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">xy'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E18:19_1_5"/><i><font color="Green" title="E33">A86</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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">xz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E19:19_1_5"/><i><font color="Green" title="E34">A87</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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="c10">xy'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c11">xz'</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="E20:19_1_5"/>
( (  not <font color="Maroon" title="c4">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="c5">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; (  not <font color="Maroon" title="c4">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="c5">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> ) &amp; (  not <font color="Maroon" title="c4">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> or  not <font color="Maroon" title="c5">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>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_5"><i><font color="Green" title="E24">A77</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_5"><i><font color="Green" title="E25">A78</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="c12">y''</font>, <font color="Maroon" title="c13">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="E22:19_1_5"/><i><font color="Green" title="E35">A88</font></i>: 
<font color="Maroon" title="c4">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="c12">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E23:19_1_5"/><i><font color="Green" title="E36">A89</font></i>: 
<font color="Maroon" title="c5">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="E24:19_1_5"/><i><font color="Green" title="E37">A90</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c12">y''</font> <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="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E25:19_1_5"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#E24:19_1_5"><i><font color="Green" title="E37">A90</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="c14">x'''</font>, <font color="Maroon" title="c15">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="E27:19_1_5"/><i><font color="Green" title="E38">A91</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'''</font>
 <b>and </b><br/><a NAME="E28:19_1_5"/><i><font color="Green" title="E39">A92</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c15">yz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E29:19_1_5"/><i><font color="Green" title="E40">A93</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c15">yz'</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#E1:19_1_5"><i><font color="Green" title="E21">A74</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><a NAME="E30:19_1_5"/><i><font color="Green" title="E41">A94</font></i>: 
<font color="Maroon" title="c10">xy'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_5"><i><font color="Green" title="E28">A81</font></i></a>, <a class="txt" href="arytm_0.html#E17:19_1_5"><i><font color="Green" title="E32">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E31:19_1_5"/><i><font color="Green" title="E42">A95</font></i>: 
<font color="Maroon" title="c7">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_5"><i><font color="Green" title="E27">A80</font></i></a>, <a class="txt" href="arytm_0.html#E22:19_1_5"><i><font color="Green" title="E35">A88</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E32:19_1_5"/><i><font color="Green" title="E43">A96</font></i>: 
<font color="Maroon" title="c9">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#E12:19_1_5"><i><font color="Green" title="E30">A83</font></i></a>, <a class="txt" href="arytm_0.html#E23:19_1_5"><i><font color="Green" title="E36">A89</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">y''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c13">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#E24:19_1_5"><i><font color="Green" title="E37">A90</font></i></a>, <a class="txt" href="arytm_0.html#E28:19_1_5"><i><font color="Green" title="E39">A92</font></i></a>, <a class="txt" href="arytm_0.html#E29:19_1_5"><i><font color="Green" title="E40">A93</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="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</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#E7:19_1_5"><i><font color="Green" title="E26">A79</font></i></a>, <a class="txt" href="arytm_0.html#E27:19_1_5"><i><font color="Green" title="E38">A91</font></i></a>, <a class="txt" href="arytm_0.html#E31:19_1_5"><i><font color="Green" title="E42">A95</font></i></a>, <a class="txt" href="arytm_0.html#E32:19_1_5"><i><font color="Green" title="E43">A96</font></i></a>, <a class="ref" href="arytm_2.html#T14" title="ARYTM_2:th.14">ARYTM_2:14</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E7:19_1_5"><i><font color="Green" title="E26">A79</font></i></a>, <a class="txt" href="arytm_0.html#E11:19_1_5"><i><font color="Green" title="E29">A82</font></i></a>, <a class="txt" href="arytm_0.html#E13:19_1_5"><i><font color="Green" title="E31">A84</font></i></a>, <a class="txt" href="arytm_0.html#E18:19_1_5"><i><font color="Green" title="E33">A86</font></i></a>, <a class="txt" href="arytm_0.html#E19:19_1_5"><i><font color="Green" title="E34">A87</font></i></a>, <a class="txt" href="arytm_0.html#E30:19_1_5"><i><font color="Green" title="E41">A94</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="19_1_6"><b>suppose </b></a><b>that </b><a NAME="E1:19_1_6"/><i><font color="Green" title="E21">A97</font></i>: 
<font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E2:19_1_6"/><i><font color="Green" title="E22">A98</font></i>: 
<font color="Maroon" title="c4">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="E3:19_1_6"/><i><font color="Green" title="E23">A99</font></i>: 
<font color="Maroon" title="c5">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"><a NAME="E4:19_1_6"/><i><font color="Green" title="E24">A100</font></i>: 
<font color="Maroon" title="c3">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#E1:19_1_6"><i><font color="Green" title="E21">A97</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">y''</font>, <font color="Maroon" title="c7">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="E6:19_1_6"/><i><font color="Green" title="E25">A101</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y''</font>
 <b>and </b><br/><a NAME="E7:19_1_6"/><i><font color="Green" title="E26">A102</font></i>: 
<font color="Maroon" title="c5">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="E8:19_1_6"/><i><font color="Green" title="E27">A103</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:19_1_6"><i><font color="Green" title="E22">A98</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_6"><i><font color="Green" title="E23">A99</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="19_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_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="E1:19_1_6_1_1"/>
( ( <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 ) or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_6_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:19_1_6_1_1_1"/><i><font color="Green" title="E28">A104</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:19_1_6_1_1_1"/><i><font color="Green" title="E29">A105</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c8">x'</font>, <font color="Maroon" title="c9">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="E4:19_1_6_1_1_1"/><i><font color="Green" title="E30">A106</font></i>: 
<font color="Maroon" title="c3">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="E5:19_1_6_1_1_1"/><i><font color="Green" title="E31">A107</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y'</font>
 <b>and </b><br/><a NAME="E6:19_1_6_1_1_1"/><i><font color="Green" title="E32">A108</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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">y'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</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="txt" href="arytm_0.html#E2:19_1_6"><i><font color="Green" title="E22">A98</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_6"><i><font color="Green" title="E24">A100</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_6_1_1_1"><i><font color="Green" title="E28">A104</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">x''</font>, <font color="Maroon" title="c11">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="E8:19_1_6_1_1_1"/><i><font color="Green" title="E33">A109</font></i>: 
<font color="Maroon" title="c3">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="c10">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_6_1_1_1"/><i><font color="Green" title="E34">A110</font></i>: 
<font color="Maroon" title="c5">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="E10:19_1_6_1_1_1"/><i><font color="Green" title="E35">A111</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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="c11">z'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c10">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#E3:19_1_6"><i><font color="Green" title="E23">A99</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_6"><i><font color="Green" title="E24">A100</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_6_1_1_1"><i><font color="Green" title="E29">A105</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><a NAME="E11:19_1_6_1_1_1"/>
(  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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;  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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#E6:19_1_6_1_1_1"><i><font color="Green" title="E32">A108</font></i></a>, <a class="txt" href="arytm_0.html#E10:19_1_6_1_1_1"><i><font color="Green" title="E35">A111</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="E12:19_1_6_1_1_1"/><b>then </b>
( (  not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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; (  not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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> ) &amp; (  not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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> or  not  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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>by </b><i><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="c12">xy'</font>, <font color="Maroon" title="c13">xz'</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="E14:19_1_6_1_1_1"/><i><font color="Green" title="E36">A112</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c12">xy'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E15:19_1_6_1_1_1"/><i><font color="Green" title="E37">A113</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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">xz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E16:19_1_6_1_1_1"/><i><font color="Green" title="E38">A114</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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="c12">xy'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c13">xz'</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="E17:19_1_6_1_1_1"/>
<font color="Maroon" title="c6">y''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">z''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:19_1_6"><i><font color="Green" title="E26">A102</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_6_1_1_1"><i><font color="Green" title="E29">A105</font></i></a>, <a class="ref" href="arytm_2.html#T6" title="ARYTM_2:th.6">ARYTM_2:6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">x'''</font>, <font color="Maroon" title="c15">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="E19:19_1_6_1_1_1"/><i><font color="Green" title="E39">A115</font></i>: 
<font color="Maroon" title="c3">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="E20:19_1_6_1_1_1"/><i><font color="Green" title="E40">A116</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yz'</font>
 <b>and </b><br/><a NAME="E21:19_1_6_1_1_1"/><i><font color="Green" title="E41">A117</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c15">yz'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</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="txt" href="arytm_0.html#E4:19_1_6"><i><font color="Green" title="E24">A100</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_6"><i><font color="Green" title="E27">A103</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><a NAME="E22:19_1_6_1_1_1"/><i><font color="Green" title="E42">A118</font></i>: 
<font color="Maroon" title="c12">xy'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:19_1_6_1_1_1"><i><font color="Green" title="E32">A108</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_6_1_1_1"><i><font color="Green" title="E36">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="E23:19_1_6_1_1_1"/><i><font color="Green" title="E43">A119</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_6_1_1_1"><i><font color="Green" title="E30">A106</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_6_1_1_1"><i><font color="Green" title="E33">A109</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E24:19_1_6_1_1_1"/>
<font color="Maroon" title="c10">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#E8:19_1_6_1_1_1"><i><font color="Green" title="E33">A109</font></i></a>, <a class="txt" href="arytm_0.html#E19:19_1_6_1_1_1"><i><font color="Green" title="E39">A115</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 href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">y'</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c11">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#E6:19_1_6"><i><font color="Green" title="E25">A101</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_6"><i><font color="Green" title="E26">A102</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_6"><i><font color="Green" title="E27">A103</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_6_1_1_1"><i><font color="Green" title="E31">A107</font></i></a>, <a class="txt" href="arytm_0.html#E9:19_1_6_1_1_1"><i><font color="Green" title="E34">A110</font></i></a>, <a class="txt" href="arytm_0.html#E20:19_1_6_1_1_1"><i><font color="Green" title="E40">A116</font></i></a>, <a class="txt" href="arytm_0.html#E21:19_1_6_1_1_1"><i><font color="Green" title="E41">A117</font></i></a>, <a class="txt" href="arytm_0.html#E23:19_1_6_1_1_1"><i><font color="Green" title="E43">A119</font></i></a>, <a class="ref" href="arytm_2.html#T14" title="ARYTM_2:th.14">ARYTM_2:14</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E10:19_1_6_1_1_1"><i><font color="Green" title="E35">A111</font></i></a>, <a class="txt" href="arytm_0.html#E15:19_1_6_1_1_1"><i><font color="Green" title="E37">A113</font></i></a>, <a class="txt" href="arytm_0.html#E16:19_1_6_1_1_1"><i><font color="Green" title="E38">A114</font></i></a>, <a class="txt" href="arytm_0.html#E22:19_1_6_1_1_1"><i><font color="Green" title="E42">A118</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="19_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_6_1_1_2"/><i><font color="Green" title="E28">A120</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
0
<b>by </b><i><a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th13</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_6_1_1_2"><i><font color="Green" title="E28">A120</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_6_1_1_2"><i><font color="Green" title="E28">A120</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:19_1_6_1_1_3"/><i><font color="Green" title="E28">A121</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th13</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th13</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_6_1_1_3"><i><font color="Green" title="E28">A121</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_6_1_1_4"><b>suppose </b></a><a NAME="E1:19_1_6_1_1_4"/><i><font color="Green" title="E28">A122</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th13</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c2">o</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th13</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_6_1_1_4"><i><font color="Green" title="E28">A122</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:19_1_6"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_7"><b>suppose </b></a><b>that </b><a NAME="E1:19_1_7"/><i><font color="Green" title="E21">A123</font></i>: 
<font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E2:19_1_7"/><i><font color="Green" title="E22">A124</font></i>: 
<font color="Maroon" title="c4">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="E3:19_1_7"/><i><font color="Green" title="E23">A125</font></i>: 
<font color="Maroon" title="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_7"/><i><font color="Green" title="E24">A126</font></i>: 
<font color="Maroon" title="c3">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#E1:19_1_7"><i><font color="Green" title="E21">A123</font></i></a></i>;<br/><a NAME="E5:19_1_7"/><i><font color="Green" title="E25">A127</font></i>: 
<font color="Maroon" title="c5">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#E3:19_1_7"><i><font color="Green" title="E23">A125</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">x''</font>, <font color="Maroon" title="c7">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="E7:19_1_7"/><i><font color="Green" title="E26">A128</font></i>: 
<font color="Maroon" title="c3">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="c6">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:19_1_7"/><i><font color="Green" title="E27">A129</font></i>: 
<font color="Maroon" title="c5">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="c7">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_7"/><i><font color="Green" title="E28">A130</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c6">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_7"><i><font color="Green" title="E24">A126</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_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="E1:19_1_7_1_1"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_7_1_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">x'</font>, <font color="Maroon" title="c9">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="E3:19_1_7_1_1_1"/><i><font color="Green" title="E29">A131</font></i>: 
<font color="Maroon" title="c3">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="E4:19_1_7_1_1_1"/><i><font color="Green" title="E30">A132</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y'</font>
 <b>and </b><br/><a NAME="E5:19_1_7_1_1_1"/><i><font color="Green" title="E31">A133</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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">y'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</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="txt" href="arytm_0.html#E2:19_1_7"><i><font color="Green" title="E22">A124</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_7"><i><font color="Green" title="E24">A126</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">y''</font>, <font color="Maroon" title="c11">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="E7:19_1_7_1_1_1"/><i><font color="Green" title="E32">A134</font></i>: 
<font color="Maroon" title="c4">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="E8:19_1_7_1_1_1"/><i><font color="Green" title="E33">A135</font></i>: 
<font color="Maroon" title="c5">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="E9:19_1_7_1_1_1"/><i><font color="Green" title="E34">A136</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</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="c11">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:19_1_7"><i><font color="Green" title="E25">A127</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_7_1_1_1"><i><font color="Green" title="E30">A132</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E10:19_1_7_1_1_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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#E5:19_1_7_1_1_1"><i><font color="Green" title="E31">A133</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="c12">xy'</font>, <font color="Maroon" title="c13">xz'</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="E12:19_1_7_1_1_1"/><i><font color="Green" title="E35">A137</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c12">xy'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E13:19_1_7_1_1_1"/><i><font color="Green" title="E36">A138</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xz'</font>
 <b>and </b><br/><a NAME="E14:19_1_7_1_1_1"/><i><font color="Green" title="E37">A139</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xz'</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">xy'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_7"><i><font color="Green" title="E28">A130</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E15:19_1_7_1_1_1"/><i><font color="Green" title="E38">A140</font></i>: 
<font color="Maroon" title="c7">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_7"><i><font color="Green" title="E27">A129</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_7_1_1_1"><i><font color="Green" title="E33">A135</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="19_1_7_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_7_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_7_1_1_1_1_1"/>
( <font color="Maroon" title="c11">z''</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="c11">z''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y''</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_7_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_7_1_1_1_1_1_1"/><i><font color="Green" title="E39">A141</font></i>: 
<font color="Maroon" title="c11">z''</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="E2:19_1_7_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E40">A142</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</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="c11">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_7_1_1_1"><i><font color="Green" title="E34">A136</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_7_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_7_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_7_1_1_1_1_1_1_1_1"/>
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_7_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_7_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E41">A143</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c14">x'''</font>, <font color="Maroon" title="c15">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="E3:19_1_7_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E42">A144</font></i>: 
<font color="Maroon" title="c3">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="E4:19_1_7_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E43">A145</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yz'</font>
 <b>and </b><br/><a NAME="E5:19_1_7_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E44">A146</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c15">yz'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</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="txt" href="arytm_0.html#E4:19_1_7"><i><font color="Green" title="E24">A126</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_7_1_1_1_1_1_1"><i><font color="Green" title="E40">A142</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><a NAME="E6:19_1_7_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E45">A147</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:19_1_7"><i><font color="Green" title="E26">A128</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_7_1_1_1"><i><font color="Green" title="E29">A131</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E7:19_1_7_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E46">A148</font></i>: 
<font color="Maroon" title="c6">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#E7:19_1_7"><i><font color="Green" title="E26">A128</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_7_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E42">A144</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E8:19_1_7_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E47">A149</font></i>: 
<font color="Maroon" title="c7">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_7"><i><font color="Green" title="E27">A129</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_7_1_1_1"><i><font color="Green" title="E33">A135</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:19_1_7_1_1_1_1_1_1_1_1_1"/>
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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/><a NAME="E10:19_1_7_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">x'''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:19_1_7_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E42">A144</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">z'</font></span>)</span> <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_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_7_1_1_1"><i><font color="Green" title="E30">A132</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_7_1_1_1"><i><font color="Green" title="E32">A134</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_7_1_1_1_1_1_1"><i><font color="Green" title="E39">A141</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_7_1_1_1_1_1_1"><i><font color="Green" title="E40">A142</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_7_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E41">A143</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_7_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E43">A145</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_7_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E44">A146</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_7_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E45">A147</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_7_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E46">A148</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_7_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E47">A149</font></i></a>, <a class="ref" href="arytm_1.html#T28" title="ARYTM_1:th.28">ARYTM_1:28</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_7"><i><font color="Green" title="E28">A130</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_7_1_1_1"><i><font color="Green" title="E31">A133</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_7_1_1_1"><i><font color="Green" title="E35">A137</font></i></a>, <a class="txt" href="arytm_0.html#E13:19_1_7_1_1_1"><i><font color="Green" title="E36">A138</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_7_1_1_1"><i><font color="Green" title="E37">A139</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_7_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E45">A147</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="19_1_7_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_7_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E41">A150</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_7_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E42">A151</font></i>: 
<font color="Maroon" title="c10">y''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_7_1_1_1_1_1_1"><i><font color="Green" title="E39">A141</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_7_1_1_1_1_1_1"><i><font color="Green" title="E40">A142</font></i></a>, <a class="ref" href="arytm_1.html#T10" title="ARYTM_1:th.10">ARYTM_1:10</a></i>;<br/><a NAME="E3:19_1_7_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E43">A152</font></i>: 
<font color="Maroon" title="c12">xy'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">y'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:19_1_7_1_1_1"><i><font color="Green" title="E31">A133</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_7_1_1_1"><i><font color="Green" title="E35">A137</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E4:19_1_7_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E44">A153</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:19_1_7"><i><font color="Green" title="E26">A128</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_7_1_1_1"><i><font color="Green" title="E29">A131</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
0
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_7_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E41">A150</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_7"><i><font color="Green" title="E28">A130</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_7_1_1_1"><i><font color="Green" title="E30">A132</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_7_1_1_1"><i><font color="Green" title="E32">A134</font></i></a>, <a class="txt" href="arytm_0.html#E13:19_1_7_1_1_1"><i><font color="Green" title="E36">A138</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_7_1_1_1"><i><font color="Green" title="E37">A139</font></i></a>, <a class="txt" href="arytm_0.html#E15:19_1_7_1_1_1"><i><font color="Green" title="E38">A140</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_7_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E42">A151</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_7_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E43">A152</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_7_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E44">A153</font></i></a>, <a class="ref" href="arytm_1.html#T18" title="ARYTM_1:th.18">ARYTM_1:18</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:19_1_7_1_1_1_1_1_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_7_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_7_1_1_1_1_1_2"/><i><font color="Green" title="E39">A154</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="c10">y''</font>
 ;<br/><div class="add"><a NAME="E2:19_1_7_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E40">A155</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c11">z''</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#E9:19_1_7_1_1_1"><i><font color="Green" title="E34">A136</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="E3:19_1_7_1_1_1_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c14">x'''</font>, <font color="Maroon" title="c15">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="E5:19_1_7_1_1_1_1_1_2"/><i><font color="Green" title="E41">A156</font></i>: 
<font color="Maroon" title="c3">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="E6:19_1_7_1_1_1_1_1_2"/><i><font color="Green" title="E42">A157</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c15">yz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:19_1_7_1_1_1_1_1_2"/><i><font color="Green" title="E43">A158</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yz'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c14">x'''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_7"><i><font color="Green" title="E24">A126</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><a NAME="E8:19_1_7_1_1_1_1_1_2"/><i><font color="Green" title="E44">A159</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:19_1_7"><i><font color="Green" title="E26">A128</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_7_1_1_1"><i><font color="Green" title="E29">A131</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:19_1_7_1_1_1_1_1_2"/><i><font color="Green" title="E45">A160</font></i>: 
<font color="Maroon" title="c6">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#E7:19_1_7"><i><font color="Green" title="E26">A128</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_7_1_1_1_1_1_2"><i><font color="Green" title="E41">A156</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
<font color="Maroon" title="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">z''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y''</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E2:19_1_7_1_1_1_1_1_2"><i><font color="Green" title="E40">A155</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_7_1_1_1_1_1_2"><i><font color="Green" title="E42">A157</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_7_1_1_1_1_1_2"><i><font color="Green" title="E43">A158</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">(<span class="default"><font color="Maroon" title="c6">x''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">z'</font></span>)</span> <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_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_7_1_1_1"><i><font color="Green" title="E30">A132</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_7_1_1_1"><i><font color="Green" title="E32">A134</font></i></a>, <a class="txt" href="arytm_0.html#E15:19_1_7_1_1_1"><i><font color="Green" title="E38">A140</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_7_1_1_1_1_1_2"><i><font color="Green" title="E39">A154</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_7_1_1_1_1_1_2"><i><font color="Green" title="E44">A159</font></i></a>, <a class="txt" href="arytm_0.html#E9:19_1_7_1_1_1_1_1_2"><i><font color="Green" title="E45">A160</font></i></a>, <a class="ref" href="arytm_1.html#T26" title="ARYTM_1:th.26">ARYTM_1:26</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_7"><i><font color="Green" title="E28">A130</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_7_1_1_1"><i><font color="Green" title="E31">A133</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_7_1_1_1"><i><font color="Green" title="E35">A137</font></i></a>, <a class="txt" href="arytm_0.html#E13:19_1_7_1_1_1"><i><font color="Green" title="E36">A138</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_7_1_1_1"><i><font color="Green" title="E37">A139</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="E17:19_1_7_1_1_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_7_1_1_2"/><i><font color="Green" title="E29">A161</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th13</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c2">o</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th13</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_7_1_1_2"><i><font color="Green" title="E29">A161</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E11:19_1_7"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_8"><b>suppose </b></a><b>that </b><a NAME="E1:19_1_8"/><i><font color="Green" title="E21">A162</font></i>: 
<font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E2:19_1_8"/><i><font color="Green" title="E22">A163</font></i>: 
<font color="Maroon" title="c5">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="E3:19_1_8"/><i><font color="Green" title="E23">A164</font></i>: 
<font color="Maroon" title="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_8"/><i><font color="Green" title="E24">A165</font></i>: 
<font color="Maroon" title="c3">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#E1:19_1_8"><i><font color="Green" title="E21">A162</font></i></a></i>;<br/><a NAME="E5:19_1_8"/><i><font color="Green" title="E25">A166</font></i>: 
<font color="Maroon" title="c4">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#E3:19_1_8"><i><font color="Green" title="E23">A164</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">x''</font>, <font color="Maroon" title="c7">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="E7:19_1_8"/><i><font color="Green" title="E26">A167</font></i>: 
<font color="Maroon" title="c3">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="c6">x''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:19_1_8"/><i><font color="Green" title="E27">A168</font></i>: 
<font color="Maroon" title="c4">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="E9:19_1_8"/><i><font color="Green" title="E28">A169</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c6">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_8"><i><font color="Green" title="E24">A165</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_8_1_1"/>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_8_1_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">x'</font>, <font color="Maroon" title="c9">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="E3:19_1_8_1_1_1"/><i><font color="Green" title="E29">A170</font></i>: 
<font color="Maroon" title="c3">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="E4:19_1_8_1_1_1"/><i><font color="Green" title="E30">A171</font></i>: 
<font color="Maroon" title="c5">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="E5:19_1_8_1_1_1"/><i><font color="Green" title="E31">A172</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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="c9">z'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</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="txt" href="arytm_0.html#E2:19_1_8"><i><font color="Green" title="E22">A163</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_8"><i><font color="Green" title="E24">A165</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">z''</font>, <font color="Maroon" title="c11">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="E7:19_1_8_1_1_1"/><i><font color="Green" title="E32">A173</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z''</font>
 <b>and </b><br/><a NAME="E8:19_1_8_1_1_1"/><i><font color="Green" title="E33">A174</font></i>: 
<font color="Maroon" title="c4">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="c11">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_8_1_1_1"/><i><font color="Green" title="E34">A175</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z''</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c11">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:19_1_8"><i><font color="Green" title="E25">A166</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_8_1_1_1"><i><font color="Green" title="E30">A171</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E10:19_1_8_1_1_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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#E5:19_1_8_1_1_1"><i><font color="Green" title="E31">A172</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="c12">xz'</font>, <font color="Maroon" title="c13">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="E12:19_1_8_1_1_1"/><i><font color="Green" title="E35">A176</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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">xz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E13:19_1_8_1_1_1"/><i><font color="Green" title="E36">A177</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="E14:19_1_8_1_1_1"/><i><font color="Green" title="E37">A178</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <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="c12">xz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_8"><i><font color="Green" title="E28">A169</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E15:19_1_8_1_1_1"/><i><font color="Green" title="E38">A179</font></i>: 
<font color="Maroon" title="c7">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_8"><i><font color="Green" title="E27">A168</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_8_1_1_1"><i><font color="Green" title="E33">A174</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="19_1_8_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_8_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_8_1_1_1_1_1"/>
( <font color="Maroon" title="c11">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">z''</font> or  not <font color="Maroon" title="c11">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">z''</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_8_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_8_1_1_1_1_1_1"/><i><font color="Green" title="E39">A180</font></i>: 
<font color="Maroon" title="c11">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">z''</font>
 ;<br/><div class="add"><a NAME="E2:19_1_8_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E40">A181</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_8_1_1_1"><i><font color="Green" title="E34">A175</font></i></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_8_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_8_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_8_1_1_1_1_1_1_1_1"/>
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_8_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_8_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E41">A182</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c14">x'''</font>, <font color="Maroon" title="c15">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="E3:19_1_8_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E42">A183</font></i>: 
<font color="Maroon" title="c3">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="E4:19_1_8_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E43">A184</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">zy'</font>
 <b>and </b><br/><a NAME="E5:19_1_8_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E44">A185</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">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">z</font>,<font color="Maroon" title="c4">y</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="c15">zy'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</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="txt" href="arytm_0.html#E4:19_1_8"><i><font color="Green" title="E24">A165</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_8_1_1_1_1_1_1"><i><font color="Green" title="E40">A181</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><a NAME="E6:19_1_8_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E45">A186</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:19_1_8"><i><font color="Green" title="E26">A167</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_8_1_1_1"><i><font color="Green" title="E29">A170</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E7:19_1_8_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E46">A187</font></i>: 
<font color="Maroon" title="c6">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#E7:19_1_8"><i><font color="Green" title="E26">A167</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_8_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E42">A183</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E8:19_1_8_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E47">A188</font></i>: 
<font color="Maroon" title="c7">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_8"><i><font color="Green" title="E27">A168</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_8_1_1_1"><i><font color="Green" title="E33">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="E9:19_1_8_1_1_1_1_1_1_1_1_1"/>
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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/><a NAME="E10:19_1_8_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">x'''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:19_1_8_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E42">A183</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">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">z</font>,<font color="Maroon" title="c4">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">y'</font></span>)</span> <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_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">z'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_8_1_1_1"><i><font color="Green" title="E30">A171</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_8_1_1_1"><i><font color="Green" title="E32">A173</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_8_1_1_1_1_1_1"><i><font color="Green" title="E39">A180</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_8_1_1_1_1_1_1"><i><font color="Green" title="E40">A181</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_8_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E41">A182</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_8_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E43">A184</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_8_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E44">A185</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_8_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E45">A186</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_8_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E46">A187</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_8_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E47">A188</font></i></a>, <a class="ref" href="arytm_1.html#T28" title="ARYTM_1:th.28">ARYTM_1:28</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_8"><i><font color="Green" title="E28">A169</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_8_1_1_1"><i><font color="Green" title="E31">A172</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_8_1_1_1"><i><font color="Green" title="E35">A176</font></i></a>, <a class="txt" href="arytm_0.html#E13:19_1_8_1_1_1"><i><font color="Green" title="E36">A177</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_8_1_1_1"><i><font color="Green" title="E37">A178</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_8_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E45">A186</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="19_1_8_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_8_1_1_1_1_1_1_1_1_2"/><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="c5">z</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_8_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E42">A190</font></i>: 
<font color="Maroon" title="c10">z''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_8_1_1_1_1_1_1"><i><font color="Green" title="E39">A180</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_8_1_1_1_1_1_1"><i><font color="Green" title="E40">A181</font></i></a>, <a class="ref" href="arytm_1.html#T10" title="ARYTM_1:th.10">ARYTM_1:10</a></i>;<br/><a NAME="E3:19_1_8_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E43">A191</font></i>: 
<font color="Maroon" title="c12">xz'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">z'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:19_1_8_1_1_1"><i><font color="Green" title="E31">A172</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_8_1_1_1"><i><font color="Green" title="E35">A176</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E4:19_1_8_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E44">A192</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:19_1_8"><i><font color="Green" title="E26">A167</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_8_1_1_1"><i><font color="Green" title="E29">A170</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">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">z</font>,<font color="Maroon" title="c4">y</font></span>)</span> = 
0
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_8_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E41">A189</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_8"><i><font color="Green" title="E28">A169</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_8_1_1_1"><i><font color="Green" title="E30">A171</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_8_1_1_1"><i><font color="Green" title="E32">A173</font></i></a>, <a class="txt" href="arytm_0.html#E13:19_1_8_1_1_1"><i><font color="Green" title="E36">A177</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_8_1_1_1"><i><font color="Green" title="E37">A178</font></i></a>, <a class="txt" href="arytm_0.html#E15:19_1_8_1_1_1"><i><font color="Green" title="E38">A179</font></i></a>, <a class="txt" href="arytm_0.html#E2:19_1_8_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E42">A190</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_8_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E43">A191</font></i></a>, <a class="txt" href="arytm_0.html#E4:19_1_8_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E44">A192</font></i></a>, <a class="ref" href="arytm_1.html#T18" title="ARYTM_1:th.18">ARYTM_1:18</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:19_1_8_1_1_1_1_1_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_8_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_8_1_1_1_1_1_2"/><i><font color="Green" title="E39">A193</font></i>: 
not <font color="Maroon" title="c11">y''</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">z''</font>
 ;<br/><div class="add"><a NAME="E2:19_1_8_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E40">A194</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">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="c11">y''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">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#E9:19_1_8_1_1_1"><i><font color="Green" title="E34">A175</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="E3:19_1_8_1_1_1_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">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/><b>then consider </b><font color="Maroon" title="c14">x'''</font>, <font color="Maroon" title="c15">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="E5:19_1_8_1_1_1_1_1_2"/><i><font color="Green" title="E41">A195</font></i>: 
<font color="Maroon" title="c3">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="E6:19_1_8_1_1_1_1_1_2"/><i><font color="Green" title="E42">A196</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">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="c15">zy'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:19_1_8_1_1_1_1_1_2"/><i><font color="Green" title="E43">A197</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">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">z</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">zy'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c14">x'''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_8"><i><font color="Green" title="E24">A165</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><a NAME="E8:19_1_8_1_1_1_1_1_2"/><i><font color="Green" title="E44">A198</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:19_1_8"><i><font color="Green" title="E26">A167</font></i></a>, <a class="txt" href="arytm_0.html#E3:19_1_8_1_1_1"><i><font color="Green" title="E29">A170</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:19_1_8_1_1_1_1_1_2"/><i><font color="Green" title="E45">A199</font></i>: 
<font color="Maroon" title="c6">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#E7:19_1_8"><i><font color="Green" title="E26">A167</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_8_1_1_1_1_1_2"><i><font color="Green" title="E41">A195</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
<font color="Maroon" title="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y''</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">z''</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E2:19_1_8_1_1_1_1_1_2"><i><font color="Green" title="E40">A194</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_8_1_1_1_1_1_2"><i><font color="Green" title="E42">A196</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_8_1_1_1_1_1_2"><i><font color="Green" title="E43">A197</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">(<span class="default"><font color="Maroon" title="c6">x''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c7">y'</font></span>)</span> <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_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c9">z'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_8_1_1_1"><i><font color="Green" title="E30">A171</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_8_1_1_1"><i><font color="Green" title="E32">A173</font></i></a>, <a class="txt" href="arytm_0.html#E15:19_1_8_1_1_1"><i><font color="Green" title="E38">A179</font></i></a>, <a class="txt" href="arytm_0.html#E1:19_1_8_1_1_1_1_1_2"><i><font color="Green" title="E39">A193</font></i></a>, <a class="txt" href="arytm_0.html#E8:19_1_8_1_1_1_1_1_2"><i><font color="Green" title="E44">A198</font></i></a>, <a class="txt" href="arytm_0.html#E9:19_1_8_1_1_1_1_1_2"><i><font color="Green" title="E45">A199</font></i></a>, <a class="ref" href="arytm_1.html#T26" title="ARYTM_1:th.26">ARYTM_1:26</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_8"><i><font color="Green" title="E28">A169</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_8_1_1_1"><i><font color="Green" title="E31">A172</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_8_1_1_1"><i><font color="Green" title="E35">A176</font></i></a>, <a class="txt" href="arytm_0.html#E13:19_1_8_1_1_1"><i><font color="Green" title="E36">A177</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_8_1_1_1"><i><font color="Green" title="E37">A178</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="E17:19_1_8_1_1_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_8_1_1_2"/><i><font color="Green" title="E29">A200</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th13</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th13</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_8_1_1_2"><i><font color="Green" title="E29">A200</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E11:19_1_8"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_9"><b>suppose </b></a><b>that </b><a NAME="E1:19_1_9"/><i><font color="Green" title="E21">A201</font></i>: 
<font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E2:19_1_9"/><i><font color="Green" title="E22">A202</font></i>: 
<font color="Maroon" title="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E3:19_1_9"/><i><font color="Green" title="E23">A203</font></i>: 
<font color="Maroon" title="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_9"/><i><font color="Green" title="E24">A204</font></i>: 
<font color="Maroon" title="c3">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#E1:19_1_9"><i><font color="Green" title="E21">A201</font></i></a></i>;<br/><a NAME="E5:19_1_9"/><i><font color="Green" title="E25">A205</font></i>: 
<font color="Maroon" title="c4">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#E2:19_1_9"><i><font color="Green" title="E22">A202</font></i></a></i>;<br/><a NAME="E6:19_1_9"/><i><font color="Green" title="E26">A206</font></i>: 
<font color="Maroon" title="c5">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#E3:19_1_9"><i><font color="Green" title="E23">A203</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">x'</font>, <font color="Maroon" title="c7">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="E8:19_1_9"/><i><font color="Green" title="E27">A207</font></i>: 
<font color="Maroon" title="c3">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="c6">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_9"/><i><font color="Green" title="E28">A208</font></i>: 
<font color="Maroon" title="c4">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="E10:19_1_9"/><i><font color="Green" title="E29">A209</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c6">x'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_9"><i><font color="Green" title="E24">A204</font></i></a>, <a class="txt" href="arytm_0.html#E5:19_1_9"><i><font color="Green" title="E25">A205</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">x''</font>, <font color="Maroon" title="c9">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="E12:19_1_9"/><i><font color="Green" title="E30">A210</font></i>: 
<font color="Maroon" title="c3">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="E13:19_1_9"/><i><font color="Green" title="E31">A211</font></i>: 
<font color="Maroon" title="c5">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="c9">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E14:19_1_9"/><i><font color="Green" title="E32">A212</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c8">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_9"><i><font color="Green" title="E24">A204</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_9"><i><font color="Green" title="E26">A206</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">xy'</font>, <font color="Maroon" title="c11">xz'</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="E16:19_1_9"/><i><font color="Green" title="E33">A213</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">xy'</font>
 <b>and </b><br/><a NAME="E17:19_1_9"/><i><font color="Green" title="E34">A214</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xz'</font>
 <b>and </b><br/><a NAME="E18:19_1_9"/><i><font color="Green" title="E35">A215</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">xy'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c11">xz'</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E10:19_1_9"><i><font color="Green" title="E29">A209</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_9"><i><font color="Green" title="E32">A212</font></i></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E19:19_1_9"/>
( (  not <font color="Maroon" title="c4">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="c5">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; (  not <font color="Maroon" title="c4">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="c5">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> ) &amp; (  not <font color="Maroon" title="c4">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> or  not <font color="Maroon" title="c5">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>by </b><i><a class="txt" href="arytm_0.html#E5:19_1_9"><i><font color="Green" title="E25">A205</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_9"><i><font color="Green" title="E26">A206</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="c12">y''</font>, <font color="Maroon" title="c13">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="E21:19_1_9"/><i><font color="Green" title="E36">A216</font></i>: 
<font color="Maroon" title="c4">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="c12">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E22:19_1_9"/><i><font color="Green" title="E37">A217</font></i>: 
<font color="Maroon" title="c5">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="E23:19_1_9"/><i><font color="Green" title="E38">A218</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c12">y''</font> <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="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></i>;<br/><a NAME="E24:19_1_9"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#E23:19_1_9"><i><font color="Green" title="E38">A218</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="c14">x'''</font>, <font color="Maroon" title="c15">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="E26:19_1_9"/><i><font color="Green" title="E39">A219</font></i>: 
<font color="Maroon" title="c3">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="E27:19_1_9"/><i><font color="Green" title="E40">A220</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c15">yz'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E28:19_1_9"/><i><font color="Green" title="E41">A221</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yz'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c14">x'''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E4:19_1_9"><i><font color="Green" title="E24">A204</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a></i>;<br/><a NAME="E29:19_1_9"/><i><font color="Green" title="E42">A222</font></i>: 
<font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E8:19_1_9"><i><font color="Green" title="E27">A207</font></i></a>, <a class="txt" href="arytm_0.html#E12:19_1_9"><i><font color="Green" title="E30">A210</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E30:19_1_9"/><i><font color="Green" title="E43">A223</font></i>: 
<font color="Maroon" title="c6">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#E8:19_1_9"><i><font color="Green" title="E27">A207</font></i></a>, <a class="txt" href="arytm_0.html#E26:19_1_9"><i><font color="Green" title="E39">A219</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E31:19_1_9"/><i><font color="Green" title="E44">A224</font></i>: 
<font color="Maroon" title="c7">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y''</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:19_1_9"><i><font color="Green" title="E28">A208</font></i></a>, <a class="txt" href="arytm_0.html#E21:19_1_9"><i><font color="Green" title="E36">A216</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E32:19_1_9"/><i><font color="Green" title="E45">A225</font></i>: 
<font color="Maroon" title="c9">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#E13:19_1_9"><i><font color="Green" title="E31">A211</font></i></a>, <a class="txt" href="arytm_0.html#E22:19_1_9"><i><font color="Green" title="E37">A217</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> = 
<font color="Maroon" title="c14">x'''</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y''</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c13">z''</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E23:19_1_9"><i><font color="Green" title="E38">A218</font></i></a>, <a class="txt" href="arytm_0.html#E27:19_1_9"><i><font color="Green" title="E40">A220</font></i></a>, <a class="txt" href="arytm_0.html#E28:19_1_9"><i><font color="Green" title="E41">A221</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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E10:19_1_9"><i><font color="Green" title="E29">A209</font></i></a>, <a class="txt" href="arytm_0.html#E14:19_1_9"><i><font color="Green" title="E32">A212</font></i></a>, <a class="txt" href="arytm_0.html#E16:19_1_9"><i><font color="Green" title="E33">A213</font></i></a>, <a class="txt" href="arytm_0.html#E17:19_1_9"><i><font color="Green" title="E34">A214</font></i></a>, <a class="txt" href="arytm_0.html#E18:19_1_9"><i><font color="Green" title="E35">A215</font></i></a>, <a class="txt" href="arytm_0.html#E29:19_1_9"><i><font color="Green" title="E42">A222</font></i></a>, <a class="txt" href="arytm_0.html#E30:19_1_9"><i><font color="Green" title="E43">A223</font></i></a>, <a class="txt" href="arytm_0.html#E31:19_1_9"><i><font color="Green" title="E44">A224</font></i></a>, <a class="txt" href="arytm_0.html#E32:19_1_9"><i><font color="Green" title="E45">A225</font></i></a>, <a class="ref" href="arytm_2.html#T14" title="ARYTM_2:th.14">ARYTM_2:14</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_10"><b>suppose </b></a><a NAME="E1:19_1_10"/><i><font color="Green" title="E21">A226</font></i>: 
( (  not <font color="Maroon" title="c3">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="c4">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="c5">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; (  not <font color="Maroon" title="c3">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="c4">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="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c3">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="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c5">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; (  not <font color="Maroon" title="c3">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="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c4">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="c5">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; (  not <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c4">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="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c5">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; (  not <font color="Maroon" title="c3">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c4">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  not <font color="Maroon" title="c5">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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 ;<br/><div class="add"><a NAME="E2:19_1_10"/><i><font color="Green" title="E22">A227</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>by </b><i><a class="ref" href="arytm_2.html#T3" title="ARYTM_2:th.3">ARYTM_2:3</a></i>;<br/> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = 
<span class="p1">(<span class="default"><a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3"><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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</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_1.html#T42" title="XBOOLE_1:th.42">XBOOLE_1:42</a></i>
<br/>.= 
<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">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3"><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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E2:19_1_10"><i><font color="Green" title="E22">A227</font></i></a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>
;<br/><b>hence </b><a NAME="E4:19_1_10"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:19_1_10"><i><font color="Green" title="E21">A226</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
