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<b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> , <font color="Maroon">f'</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:20_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <font color="Maroon">f</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0 &amp; ( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> implies <font color="Maroon">f</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>and </b><br/><a NAME="E2:20_1_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <font color="Maroon">f'</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0 &amp; ( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> implies <font color="Maroon">f'</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> 1 ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_2_1"/>
( <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_1"><b>suppose </b></a><a NAME="E1:20_1_2_1_1"/>
<font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:20_1_2_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><b>hence </b><font color="Maroon">f</font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="partfun1.html#T57">PARTFUN1:57</a></i>
<br/>.= 
<font color="Maroon">f'</font>
<b>by </b><i>, <a class="ref" href="partfun1.html#T57">PARTFUN1:57</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_2"><b>suppose </b></a><a NAME="E1:20_1_2_1_2"/>
not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_2_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_2_1_2_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>thus </b><font color="Maroon">f</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
0
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f'</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_2_1_2_1_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>thus </b><font color="Maroon">f</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
1
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f'</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:20_1_2_1_2_1"/>
<font color="Maroon">f</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_2_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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