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<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">o</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">o</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">o</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x9</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y9</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></span>;<br/><a NAME="E3:17_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x9</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c6">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#T4" title="ARYTM_2:th.4">ARYTM_2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:17_1_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">o</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_2"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">o</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:17_1_2"/>
 not <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_1_2"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">o</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:17_1_2"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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