<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">z</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">z</font>] </span>}</span> ;<br/>
<a NAME="E1:2_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">z</font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_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="E1:2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">z</font>] </span>}</span> 
 ;<br/>

<a NAME="E2:2_1_1"/><b>then </b>
 ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">z</font>] )
 
;<br/>
<b>hence </b><a NAME="E3:2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:2_1"/>
<span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">z</font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_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="E1:2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">z</font>] </span>}</span> 
 ;<br/>

<a NAME="E2:2_1_2"/><b>then </b>
 ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">z</font>] )
 
;<br/>
<b>hence </b><a NAME="E3:2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">z</font>] </span>}</span> , <font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">z</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<a NAME="E4:2_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</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">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:2_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:2_1_3"/><b>then </b>
(  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">z</font>] ) &amp;  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">z</font>] ) )
 
;<br/>
<b>hence </b><a NAME="E3:2_1_3"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">z</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E6:2_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</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">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>( <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font> )
 <b>by </b><i><a class="ref" href="axioms.html#T26">AXIOMS:26</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E7:2_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">y</font>] )
 ;<br/>

<a NAME="E8:2_1"/>
( <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E9:2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:2_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>


</div>
