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<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">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">x</font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</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">E13</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
0
<b>by </b><i/>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">o</font>,<font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">o</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </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">E16</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:19_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:19_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E5:19_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E6:19_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E7:19_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E9:19_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E10:19_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>and </b><br/><a NAME="E11:19_1_2"/><i><font color="Green">E41</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E2:19_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">xy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">xz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E13:19_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xy'</font>
 <b>and </b><br/><a NAME="E14:19_1_2"/><i><font color="Green">E44</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xz'</font>
 <b>and </b><br/><a NAME="E15:19_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xy'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">xz'</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, </i>;<br/><b>consider </b><font color="Maroon">y''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E17:19_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>and </b><br/><a NAME="E18:19_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font>
 <b>and </b><br/><a NAME="E19:19_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">z''</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E2:19_1_2"><i><font color="Green">E23</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">yz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E21:19_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>and </b><br/><a NAME="E22:19_1_2"/><i><font color="Green">E54</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yz'</font>
 <b>and </b><br/><a NAME="E23:19_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">yz'</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E5:19_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>thus </b><a NAME="E24:19_1_2"/>
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:19_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , , <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, , <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , <a class="txt" href="arytm_0.html#E5:19_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="arytm_0.html#E6:19_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="arytm_0.html#E7:19_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="arytm_0.html#E9:19_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="arytm_2.html#T14">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">E57</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>and </b><br/><a NAME="E2:19_1_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:19_1_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E6:19_1_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E7:19_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E8:19_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E10:19_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E11:19_1_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">z'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E12:19_1_3"/><i><font color="Green">E66</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">y''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E14:19_1_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>and </b><br/><a NAME="E15:19_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">z''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E16:19_1_3"/><i><font color="Green">E69</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">z''</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E17:19_1_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E18:19_1_3"/>
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">xy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">xz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E20:19_1_3"/><i><font color="Green">E71</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xy'</font>
 <b>and </b><br/><a NAME="E21:19_1_3"/><i><font color="Green">E72</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">xz'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E22:19_1_3"/><i><font color="Green">E73</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xy'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">xz'</font>
 <b>by </b><i>, </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">z''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">y''</font> or  not <font color="Maroon">z''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">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">E76</font></i>: 
<font color="Maroon">z''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">y''</font>
 ;<br/><div class="add"><a NAME="E2:19_1_3_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">z''</font>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">yz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E4:19_1_3_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>and </b><br/><a NAME="E5:19_1_3_1_1_1"/><i><font color="Green">E79</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yz'</font>
 <b>and </b><br/><a NAME="E6:19_1_3_1_1_1"/><i><font color="Green">E80</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">yz'</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_1.html#K2">-</a> <span class="p1">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , , , , <a class="ref" href="arytm_1.html#T26">ARYTM_1:26</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="zfmisc_1.html#T33">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">E81</font></i>: 
not <font color="Maroon">z''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">y''</font>
 ;<br/><div class="add"><a NAME="E2:19_1_3_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">z''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">y''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">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" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">yz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E5:19_1_3_1_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>and </b><br/><a NAME="E6:19_1_3_1_1_2"/><i><font color="Green">E84</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">yz'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:19_1_3_1_1_2"/><i><font color="Green">E85</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">yz'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <span class="p3">(<span class="default"><font color="Maroon">z''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">y''</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_1.html#K2">-</a> <span class="p1">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , <a class="ref" href="arytm_1.html#T27">ARYTM_1:27</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="zfmisc_1.html#T33">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" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">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">E86</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>and </b><br/><a NAME="E2:19_1_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:19_1_4"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_4"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E6:19_1_4"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E7:19_1_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>and </b><br/><a NAME="E8:19_1_4"/><i><font color="Green">E93</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E10:19_1_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E11:19_1_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">y'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E12:19_1_4"/><i><font color="Green">E96</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">z''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E14:19_1_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font>
 <b>and </b><br/><a NAME="E15:19_1_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">y''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E16:19_1_4"/><i><font color="Green">E99</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">y''</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E17:19_1_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E18:19_1_4"/>
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">xz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">xy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E20:19_1_4"/><i><font color="Green">E101</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xz'</font>
 <b>and </b><br/><a NAME="E21:19_1_4"/><i><font color="Green">E102</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">xy'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E22:19_1_4"/><i><font color="Green">E104</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xz'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">xy'</font>
 <b>by </b><i>, </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">y''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z''</font> or  not <font color="Maroon">y''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">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">E105</font></i>: 
<font color="Maroon">y''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z''</font>
 ;<br/><div class="add"><a NAME="E2:19_1_4_1_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">y''</font>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">zy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E4:19_1_4_1_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>and </b><br/><a NAME="E5:19_1_4_1_1_1"/><i><font color="Green">E108</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">zy'</font>
 <b>and </b><br/><a NAME="E6:19_1_4_1_1_1"/><i><font color="Green">E109</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">zy'</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span> <a href="arytm_1.html#K2">-</a> <span class="p1">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , , , , <a class="ref" href="arytm_1.html#T26">ARYTM_1:26</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="zfmisc_1.html#T33">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">E110</font></i>: 
not <font color="Maroon">y''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z''</font>
 ;<br/><div class="add"><a NAME="E2:19_1_4_1_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">y''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">z''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">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" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">zy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E5:19_1_4_1_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>and </b><br/><a NAME="E6:19_1_4_1_1_2"/><i><font color="Green">E113</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">zy'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:19_1_4_1_1_2"/><i><font color="Green">E114</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">zy'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <span class="p3">(<span class="default"><font color="Maroon">y''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">z''</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span> <a href="arytm_1.html#K2">-</a> <span class="p1">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , <a class="ref" href="arytm_1.html#T27">ARYTM_1:27</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="zfmisc_1.html#T33">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" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">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">E115</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>and </b><br/><a NAME="E2:19_1_5"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E3:19_1_5"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_5"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:19_1_5"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E7:19_1_5"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E8:19_1_5"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">y'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_5"/><i><font color="Green">E122</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E11:19_1_5"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E12:19_1_5"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">z'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E13:19_1_5"/><i><font color="Green">E125</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E14:19_1_5"/>
(  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E15:19_1_5"/><b>then </b>
( (  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) &amp; (  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then consider </b><font color="Maroon">xy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">xz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E17:19_1_5"/><i><font color="Green">E126</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">xy'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E18:19_1_5"/><i><font color="Green">E127</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">xz'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E19:19_1_5"/><i><font color="Green">E128</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">xy'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">xz'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E20:19_1_5"/>
( (  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) &amp; (  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) )
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then consider </b><font color="Maroon">y''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E22:19_1_5"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">y''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E23:19_1_5"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">z''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E24:19_1_5"/><i><font color="Green">E131</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">y''</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">z''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E25:19_1_5"/>
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">yz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E27:19_1_5"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>and </b><br/><a NAME="E28:19_1_5"/><i><font color="Green">E133</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">yz'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E29:19_1_5"/><i><font color="Green">E134</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">yz'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E30:19_1_5"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">xy'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E31:19_1_5"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E32:19_1_5"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <span class="p3">(<span class="default"><font color="Maroon">y''</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">z''</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_2.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="arytm_2.html#T14">ARYTM_2:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , <a class="ref" href="zfmisc_1.html#T33">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">E162</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E2:19_1_6"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:19_1_6"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><a NAME="E4:19_1_6"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">y''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E6:19_1_6"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>and </b><br/><a NAME="E7:19_1_6"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font>
 <b>and </b><br/><a NAME="E8:19_1_6"/><i><font color="Green">E168</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">z''</font>
 <b>by </b><i>, , </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">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</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">E169</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:19_1_6_1_1_1"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E4:19_1_6_1_1_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:19_1_6_1_1_1"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E6:19_1_6_1_1_1"/><i><font color="Green">E173</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E8:19_1_6_1_1_1"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_6_1_1_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>and </b><br/><a NAME="E10:19_1_6_1_1_1"/><i><font color="Green">E176</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">z'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E11:19_1_6_1_1_1"/>
(  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">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" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) &amp; (  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then consider </b><font color="Maroon">xy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">xz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E14:19_1_6_1_1_1"/><i><font color="Green">E177</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">xy'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E15:19_1_6_1_1_1"/><i><font color="Green">E178</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">xz'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E16:19_1_6_1_1_1"/><i><font color="Green">E179</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">xy'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">xz'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E17:19_1_6_1_1_1"/>
<font color="Maroon">y''</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">z''</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, <a class="ref" href="arytm_2.html#T6">ARYTM_2:6</a></i>;<br/><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">yz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E19:19_1_6_1_1_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E20:19_1_6_1_1_1"/><i><font color="Green">E181</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yz'</font>
 <b>and </b><br/><a NAME="E21:19_1_6_1_1_1"/><i><font color="Green">E182</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">yz'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E22:19_1_6_1_1_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">xy'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E23:19_1_6_1_1_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E24:19_1_6_1_1_1"/>
<font color="Maroon">x''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_2.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , , <a class="ref" href="arytm_2.html#T14">ARYTM_2:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="zfmisc_1.html#T33">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">E185</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
0
<b>by </b><i/>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">o</font>,<font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">o</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </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">E186</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </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">E187</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </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" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">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">E188</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E2:19_1_7"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:19_1_7"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_7"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:19_1_7"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E7:19_1_7"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E8:19_1_7"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">z'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_7"/><i><font color="Green">E195</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x''</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></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">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</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">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E3:19_1_7_1_1_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:19_1_7_1_1_1"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E5:19_1_7_1_1_1"/><i><font color="Green">E198</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">y''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E7:19_1_7_1_1_1"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>and </b><br/><a NAME="E8:19_1_7_1_1_1"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">z''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_7_1_1_1"/><i><font color="Green">E201</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">z''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, , </i>;<br/><a NAME="E10:19_1_7_1_1_1"/>
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">xy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">xz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E12:19_1_7_1_1_1"/><i><font color="Green">E202</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">xy'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E13:19_1_7_1_1_1"/><i><font color="Green">E203</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xz'</font>
 <b>and </b><br/><a NAME="E14:19_1_7_1_1_1"/><i><font color="Green">E204</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xz'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">xy'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E15:19_1_7_1_1_1"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">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">z''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">y''</font> or  not <font color="Maroon">z''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">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">E206</font></i>: 
<font color="Maroon">z''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">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">E207</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">z''</font>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">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" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</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">E208</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">yz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">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">E209</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'''</font></span><a href="tarski.html#K4">]</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">E210</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E211</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">yz'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E6:19_1_7_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T33">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">E218</font></i>: 
<font color="Maroon">x''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="zfmisc_1.html#T33">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">E219</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">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">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a>, <a class="ref" href="xboole_0.html#D4">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">x'''</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span> <a href="arytm_1.html#K2">-</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i>, , , , , , , , , , <a class="ref" href="arytm_1.html#T28">ARYTM_1:28</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , , <a class="ref" href="zfmisc_1.html#T33">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">E220</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</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">E221</font></i>: 
<font color="Maroon">y''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font>
 <b>by </b><i>, , <a class="ref" href="arytm_1.html#T10">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">E222</font></i>: 
<font color="Maroon">xy'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , <a class="ref" href="zfmisc_1.html#T33">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">E223</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
0
<b>by </b><i>, </i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, , , , , , , , , <a class="ref" href="arytm_1.html#T18">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" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">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">E224</font></i>: 
not <font color="Maroon">z''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">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">E225</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">z''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">y''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">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" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">yz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E5:19_1_7_1_1_1_1_1_2"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:19_1_7_1_1_1_1_1_2"/><i><font color="Green">E227</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">yz'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:19_1_7_1_1_1_1_1_2"/><i><font color="Green">E228</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">yz'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'''</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E8:19_1_7_1_1_1_1_1_2"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:19_1_7_1_1_1_1_1_2"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">x''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <span class="p1">(<span class="default"><font color="Maroon">z''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">y''</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span> <a href="arytm_1.html#K2">-</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i>, , , , , , <a class="ref" href="arytm_1.html#T26">ARYTM_1:26</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , <a class="ref" href="zfmisc_1.html#T33">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" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">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">E231</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T8" target="_self">Th8</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" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">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">E232</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E2:19_1_8"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:19_1_8"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_8"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:19_1_8"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E7:19_1_8"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E8:19_1_8"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">y'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_8"/><i><font color="Green">E239</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x''</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></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">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">z</font> <a href="hidden.html#R1">=</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">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E3:19_1_8_1_1_1"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:19_1_8_1_1_1"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>and </b><br/><a NAME="E5:19_1_8_1_1_1"/><i><font color="Green">E242</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">z'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">z''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E7:19_1_8_1_1_1"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font>
 <b>and </b><br/><a NAME="E8:19_1_8_1_1_1"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">y''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_8_1_1_1"/><i><font color="Green">E245</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">y''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, , </i>;<br/><a NAME="E10:19_1_8_1_1_1"/>
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">xz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">xy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E12:19_1_8_1_1_1"/><i><font color="Green">E246</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">xz'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E13:19_1_8_1_1_1"/><i><font color="Green">E247</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xy'</font>
 <b>and </b><br/><a NAME="E14:19_1_8_1_1_1"/><i><font color="Green">E248</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xy'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">xz'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E15:19_1_8_1_1_1"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">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">y''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z''</font> or  not <font color="Maroon">y''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">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">E250</font></i>: 
<font color="Maroon">y''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">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">E251</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">y''</font>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">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" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</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">E252</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">zy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">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">E253</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'''</font></span><a href="tarski.html#K4">]</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">E254</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E255</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">zy'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E6:19_1_8_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T33">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">E257</font></i>: 
<font color="Maroon">x''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="zfmisc_1.html#T33">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">E258</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">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">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a>, <a class="ref" href="xboole_0.html#D4">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">x'''</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_1.html#K2">-</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span>
<b>by </b><i>, , , , , , , , , , <a class="ref" href="arytm_1.html#T28">ARYTM_1:28</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , , <a class="ref" href="zfmisc_1.html#T33">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">E259</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</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">E260</font></i>: 
<font color="Maroon">z''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>by </b><i>, , <a class="ref" href="arytm_1.html#T10">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">E261</font></i>: 
<font color="Maroon">xz'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , <a class="ref" href="zfmisc_1.html#T33">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">E262</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> = 
0
<b>by </b><i>, </i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , , , , , , , , <a class="ref" href="arytm_1.html#T18">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" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">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">E263</font></i>: 
not <font color="Maroon">y''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">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">E264</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">y''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">z''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">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" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">zy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E5:19_1_8_1_1_1_1_1_2"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:19_1_8_1_1_1_1_1_2"/><i><font color="Green">E266</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">zy'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:19_1_8_1_1_1_1_1_2"/><i><font color="Green">E267</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">zy'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'''</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E8:19_1_8_1_1_1_1_1_2"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, <a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:19_1_8_1_1_1_1_1_2"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">x''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <span class="p1">(<span class="default"><font color="Maroon">y''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">z''</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_1.html#K2">-</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">z'</font></span>)</span>
<b>by </b><i>, , , , , , <a class="ref" href="arytm_1.html#T26">ARYTM_1:26</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , , , <a class="ref" href="zfmisc_1.html#T33">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" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">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">E270</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T8" target="_self">Th8</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" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">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">E271</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E2:19_1_9"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E3:19_1_9"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E4:19_1_9"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:19_1_9"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:19_1_9"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E8:19_1_9"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:19_1_9"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">y'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E10:19_1_9"/><i><font color="Green">E279</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E12:19_1_9"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E13:19_1_9"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">z'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E14:19_1_9"/><i><font color="Green">E282</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x''</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">xy'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">xz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E16:19_1_9"/><i><font color="Green">E283</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xy'</font>
 <b>and </b><br/><a NAME="E17:19_1_9"/><i><font color="Green">E284</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xz'</font>
 <b>and </b><br/><a NAME="E18:19_1_9"/><i><font color="Green">E285</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xy'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">xz'</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, , </i>;<br/><a NAME="E19:19_1_9"/>
( (  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) &amp; (  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) )
 <b>by </b><i><a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then consider </b><font color="Maroon">y''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E21:19_1_9"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">y''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E22:19_1_9"/><i><font color="Green">E287</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">z''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E23:19_1_9"/><i><font color="Green">E288</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">y''</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">z''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E24:19_1_9"/>
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">x'''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">yz'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E26:19_1_9"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E27:19_1_9"/><i><font color="Green">E290</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">yz'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E28:19_1_9"/><i><font color="Green">E291</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">yz'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'''</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E29:19_1_9"/><i><font color="Green">E292</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E30:19_1_9"/><i><font color="Green">E293</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E31:19_1_9"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E32:19_1_9"/><i><font color="Green">E295</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2" target="_self">Def2</a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x'''</font> <a href="arytm_2.html#K8">*'</a> <span class="p1">(<span class="default"><font color="Maroon">y''</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">z''</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T8" target="_self">Th8</a>, , , , , , , , , <a class="ref" href="arytm_2.html#T14">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">E296</font></i>: 
( (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/><div class="add"><a NAME="E2:19_1_10"/><i><font color="Green">E297</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>by </b><i><a class="ref" href="arytm_2.html#T3">ARYTM_2:3</a></i>;<br/> <a href="numbers.html#K1">REAL</a>  = 
<span class="p1">(<span class="default"><a href="arytm_2.html#K2">REAL+</a>  <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a>, <a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
<br/>.= 
<a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>
;<br/><b>hence </b><a NAME="E4:19_1_10"/>
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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