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<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</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> ;<br/>





<b>assume </b><a NAME="E1:70"/><i><font color="Green" title="E61">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c6">z</font>
 ;<br/>

<b>assume </b><a NAME="E2:70"/><i><font color="Green" title="E62">A2</font></i>: 
not <font color="Maroon" title="c6">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">y0</font> being    <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> <b> such that </b><br/><a NAME="E4:70"/><i><font color="Green" title="E63">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T10" target="_self" title="ARYTM_2:th.10">Th10</a></i>;<br/>
<a NAME="E5:70"/>
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">zz</font> =  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</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>  <b>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self" title="ARYTM_2:th.1">Th1</a></i>;<br/>
<a NAME="E7:70"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:70"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="arytm_2.html#E4:70"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="arytm_2.html#T7" target="_self" title="ARYTM_2:th.7">Th7</a></i>;<br/>
<a NAME="E8:70"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>;<br/>
<a NAME="E9:70"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#T12" target="_self" title="ARYTM_2:th.12">Th12</a></i>;<br/>
<a NAME="E10:70"/><b>then </b>
<font color="Maroon" title="c7">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#T6" target="_self" title="ARYTM_2:th.6">Th6</a></i>;<br/>
<a NAME="E11:70"/><b>then </b>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E4:70"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E12:70"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:70"><i><font color="Green" title="E62">A2</font></i></a></i>;<br/>


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