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

<b>let </b><font color="Maroon" title="c21">F</font> be   <a href="realset2.html#NM2" title="REALSET2:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c22">a</font>, <font color="Maroon" title="c23">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font>;<br/>



<a NAME="E1:57"/><i><font color="Green" title="E75">A1</font></i>: 
( <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font> or <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font> or ( <font color="Maroon" title="c22">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="realset1.html#K5" title="REALSET1:func.5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font></span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span> &amp; <font color="Maroon" title="c23">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="realset1.html#K5" title="REALSET1:func.5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font></span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E2:57"/><i><font color="Green" title="E76">A2</font></i>: 
( <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font> implies <span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c22">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green" title="E77">A3</font></i>: 
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font> = 
 <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font>
<b>by </b><i><a class="ref" href="realset2.html#T18" target="_self" title="REALSET2:th.18">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c22">a</font>
<b>by </b><i><a class="txt" href="realset2.html#E1:57_1"><i><font color="Green" title="E77">A3</font></i></a>, <a class="ref" href="realset2.html#T17" target="_self" title="REALSET2:th.17">Th17</a></i>
;<br/>
<b>hence </b><a NAME="E3:57_1"/>
<span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c22">a</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:57"/>
( <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font> implies <span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c22">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_2"/><i><font color="Green" title="E77">A4</font></i>: 
<font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font> = 
 <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font>
<b>by </b><i><a class="ref" href="realset2.html#T17" target="_self" title="REALSET2:th.17">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c22">a</font>
<b>by </b><i><a class="txt" href="realset2.html#E1:57_2"><i><font color="Green" title="E77">A4</font></i></a>, <a class="ref" href="realset2.html#T18" target="_self" title="REALSET2:th.18">Th18</a></i>
;<br/>
<b>hence </b><a NAME="E3:57_2"/>
<span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c22">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:57"/>
<span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c22">a</font>
 <b>by </b><i><a class="txt" href="realset2.html#E1:57"><i><font color="Green" title="E75">A1</font></i></a>, <a class="txt" href="realset2.html#E2:57"><i><font color="Green" title="E76">A2</font></i></a>, <a class="ref" href="realset2.html#T9" target="_self" title="REALSET2:th.9">Th9</a></i>;<br/>


</div>
