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<div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="measure6.html#K6">+</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:31_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:31_1_1_1_1"/>
 ex <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Olive">z</font> )
 
<b>by </b><i>, , <a class="ref" href="measure6.html#D6">MEASURE6:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:31_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E3:31_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">IT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:31_1_1_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>
<b>thus </b><a NAME="E4:31_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 <b>by </b><i>, , <a class="ref" href="measure6.html#D6">MEASURE6:def 6</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:31_1_1_1"/>
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:31_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Olive">r3</font></span>)</span> where <font color="Olive">r3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="real_1.html#NM1">Real</a>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 ;<br/><b>then consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:31_1_1_2_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:31_1_1_2_1"/>
 ex <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Olive">z</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon">z</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E2:31_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <font color="Maroon">z</font> )
 ;<br/><b>thus </b><a NAME="E3:31_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1_1"/>
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="measure6.html#K6">+</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="measure6.html#D6">MEASURE6:def 6</a></i>;<br/>


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