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<b>let </b><font color="Maroon" title="c3">IT</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1_1_1"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E2:31_1_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c3">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="c4">x</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><br/>

<b>assume </b><a NAME="E1:31_1_1_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E3:31_1_1_1_1"/>
 ex <font color="Olive" title="b1">z</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">z</font> )
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:31_1_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E1:31_1_1_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="measure6.html#D6" title="MEASURE6:def.6">MEASURE6:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:31_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">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" title="c4">x</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><br/>

<b>assume </b><a NAME="E1:31_1_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:31_1_1_1_2"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">r3</font> &amp; <font color="Maroon" title="c5">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>
<b>thus </b><a NAME="E4:31_1_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:31_1_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E3:31_1_1_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="measure6.html#D6" title="MEASURE6:def.6">MEASURE6:def 6</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:31_1_1_1"/>
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:31_1_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="E21">A5</font></i>: 
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="c4">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1_1_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:31_1_1_2_1"/><i><font color="Green" title="E22">A6</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">r3</font> &amp; <font color="Maroon" title="c5">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:31_1_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/><b>thus </b><a NAME="E4:31_1_1_2_1"/>
 ex <font color="Olive" title="b1">z</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">z</font> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:31_1_1_2_1"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon" title="c5">z</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E3:31_1_1_2"/><i><font color="Green" title="E22">A7</font></i>: 
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">z</font> )
 ;<br/><b>thus </b><a NAME="E4:31_1_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:31_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="pscomp_1.html#E3:31_1_1_2"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1_1"/>
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="measure6.html#D6" title="MEASURE6:def.6">MEASURE6:def 6</a></i>;<br/>


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