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<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font></span><br/><b>let </b><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font></span>)</span> or <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font> )</span><br/>



<b>assume </b><a NAME="E1:36"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">r0</font>, <font color="Maroon" title="c8">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> <b> such that </b><br/><a NAME="E3:36"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">s0</font>
 <b>and </b><br/><a NAME="E4:36"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c7">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E5:36"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c8">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c10">s1</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> <b> such that </b><br/><a NAME="E7:36"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c8">s0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r1</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c10">s1</font>
 <b>and </b><br/><a NAME="E8:36"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E9:36"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 <b>by </b><i><a class="txt" href="#E5:36"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/>
<a NAME="E10:36"/><i><font color="Green" title="E40">A7</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c10">s1</font>
 
<b>by </b><i><a class="txt" href="#E3:36"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E7:36"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></i>;<br/>
<a NAME="E11:36"/>
<font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="#E4:36"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E8:36"><i><font color="Green" title="E38">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:36"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <font color="Maroon" title="c5">C</font>
 <b>by </b><i><a class="txt" href="#E9:36"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="#E10:36"><i><font color="Green" title="E40">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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