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<b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">f'</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:20_1_2"/><i><font color="Green" title="E18">A7</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <b>and </b><br/><a NAME="E2:20_1_2"/><i><font color="Green" title="E19">A8</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Maroon" title="c3">f'</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c3">f'</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_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_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_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_2_1_1_1_1"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="metric_1.html#E1:20_1_2"><i><font color="Green" title="E18">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f'</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="txt" href="metric_1.html#E2:20_1_2"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="metric_1.html#E1:20_1_2_1_1_1_1"><i><font color="Green" title="E20">A10</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_2_1_1_1_2"/><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> = 
1
<b>by </b><i><a class="txt" href="metric_1.html#E1:20_1_2"><i><font color="Green" title="E18">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f'</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="txt" href="metric_1.html#E2:20_1_2"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="metric_1.html#E1:20_1_2_1_1_1_2"><i><font color="Green" title="E20">A11</font></i></a></i>
;<br/><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"/>
<font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f'</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:20_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f'</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


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