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<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <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> ; <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="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:36"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_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="E1:36_1"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  or <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><span class="kw">suppose </span></a><a NAME="E1:36_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:36_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</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="36_1_2"><span class="kw">suppose </span></a><a NAME="E1:36_1_2"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span><br/><div class="add"><a NAME="E2:36_1_2"/><span class="kw">then </span>
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_2.html#E19"><span class="lab"><font color="Green" title="E13">Lm10</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">s0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E4:36_1_2"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c5">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/><span class="kw">assume </span><a NAME="E5:36_1_2"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</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/><a NAME="E6:36_1_2"/><span class="kw">then </span>
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_2.html#E19"><span class="lab"><font color="Green" title="E13">Lm10</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">r0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E8:36_1_2"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon" title="c6">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/><a NAME="E9:36_1_2"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<font color="Maroon" title="c6">r0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c5">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36_1_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E4:36_1_2"><span class="lab"><font color="Green" title="E35">A3</font></span></a></span>;<br/><a NAME="E10:36_1_2"/>
 <a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E1:36_1_2"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E5:36_1_2"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:36_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_1_2"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="arytm_2.html#E20"><span class="lab"><font color="Green" title="E14">Lm11</font></span></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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