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<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E17</font></i>: 
0 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">x</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <a href="arytm_2.html#K2">REAL+</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E4:8"/><i><font color="Green">E31</font></i>: 
0 <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="arytm_2.html#T21">ARYTM_2:21</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:8"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:8"/>
for <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>   st <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Olive">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Olive">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:8_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:8_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:8_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="arytm_0.html#T1">ARYTM_0:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E7:8_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E8:8_1"/><i><font color="Green">E42</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>and </b><br/><a NAME="E9:8_1"/><i><font color="Green">E43</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a>, <a class="ref" href="arytm_2.html#T21">ARYTM_2:21</a></i>;<br/>
<a NAME="E10:8_1"/>
<font color="Maroon">xx</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:8_1"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x</font>,1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:8_1"/>
<font color="Maroon">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="txt" href="axioms.html#E1"><i><font color="Green">Lemma13</font></i></a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:8"/><b>then </b>
 <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="arytm_2.html#T18">ARYTM_2:18</a></i>;<br/>
<b>hence </b><a NAME="E8:8"/>
 <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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