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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</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="E1:142"/><span class="lab"><font color="Green" title="E65">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</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="E2:142"/>
(  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or  not <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E3:142"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A2</font></span>: 
(  not <font color="Maroon" title="c1">x</font> is  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  or  not <font color="Maroon" title="c2">y</font> is  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:142"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_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="E1:142_1"/>
( ( <font color="Maroon" title="c1">x</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  &amp; <font color="Maroon" title="c2">y</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  ) or ( <font color="Maroon" title="c1">x</font> is  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  &amp; <font color="Maroon" title="c2">y</font> is  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  ) or ( <font color="Maroon" title="c1">x</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  &amp; <font color="Maroon" title="c2">y</font> is  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  ) or ( <font color="Maroon" title="c1">x</font> is  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  &amp; <font color="Maroon" title="c2">y</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:142"><span class="lab"><font color="Green" title="E67">A3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1"><span class="kw">suppose </span></a><a NAME="E1:142_1_1"/>
( ( <font color="Maroon" title="c1">x</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  &amp; <font color="Maroon" title="c2">y</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  ) or ( <font color="Maroon" title="c1">x</font> is  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  &amp; <font color="Maroon" title="c2">y</font> is  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:142_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E3:142"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="xxreal_3.html#D5" target="_self" title="XXREAL_3:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="142_1_2"><span class="kw">suppose </span></a><a NAME="E1:142_1_2"/>
( ( <font color="Maroon" title="c1">x</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  &amp; <font color="Maroon" title="c2">y</font> is  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  ) or ( <font color="Maroon" title="c1">x</font> is  <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  &amp; <font color="Maroon" title="c2">y</font> is  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:142_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E3:142"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="xxreal_3.html#D5" target="_self" title="XXREAL_3:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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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