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

<b>let </b><font color="Maroon">F</font> be   <a href="realset2.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2">suppf</a> <font color="Maroon">F</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2">suppf</a> <font color="Maroon">F</font>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2">suppf</a> <font color="Maroon">F</font>, <font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2">suppf</a> <font color="Maroon">F</font>;<br/>



<a NAME="E1:7"/><i><font color="Green">E15</font></i>: 
( <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span><a href="tarski.html#K4">]</a></span> implies <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<i><font color="Green">E22</font></i>: <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="realset2.html#K4">ndf</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#T6">REALSET2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#D14">REALSET2:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#T5">REALSET2:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span>)</span>,<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="realset2.html#T4">REALSET2:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#T5">REALSET2:5</a></i>
;<br/>
<font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><a href="realset2.html#K4">ndf</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#T6">REALSET2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#D14">REALSET2:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span>,<font color="Maroon">d</font></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#T5">REALSET2:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span>)</span>,<font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="realset2.html#T4">REALSET2:4</a></i>
;<br/>
<b>hence </b><a NAME="E4:7_1"/>
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="realset3.html#T2" target="_self">Th2</a>, <a class="ref" href="realset2.html#T4">REALSET2:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:7"/>
( <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> implies <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_2"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 ;<br/>

<font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="realset2.html#K4">ndf</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#T6">REALSET2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#D14">REALSET2:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i>, <a class="ref" href="realset2.html#T4">REALSET2:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#T4">REALSET2:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span>)</span>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="realset2.html#T5">REALSET2:5</a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#T4">REALSET2:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="realset2.html#K4">ndf</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#D14">REALSET2:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i><a class="ref" href="realset2.html#T6">REALSET2:6</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7"/>
( <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K7">compf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span><a href="tarski.html#K4">]</a></span> iff <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3">odf</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="realset3.html#T1" target="_self">Th1</a></i>;<br/>


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