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<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/>





<b>assume </b><a NAME="E1:70"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font>
 ;<br/>

<b>assume </b><a NAME="E2:70"/><i><font color="Green">E20</font></i>: 
not <font color="Maroon">z</font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E4:70"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:70"/>
 <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">zz</font> =  <a href="arytm_3.html#K13">{}</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  <b>by </b><i/>;<br/>
<a NAME="E7:70"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="arytm_2.html#D5" target="_self">Def5</a>, <a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, </i>;<br/>
<a NAME="E8:70"/><b>then </b>
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">zz</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y0</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:70"/><b>then </b>
<font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y0</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E10:70"/><b>then </b>
<font color="Maroon">y0</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:70"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E12:70"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a></i>;<br/>


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