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<b>let </b><font color="Maroon">F1</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> , <font color="Maroon">F2</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </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="E1:24_1_2"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">F1</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>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Olive">y</font></span>)</span>
 <b>and </b><br/><a NAME="E2:24_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">F2</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>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K5">-</a> <font color="Olive">y</font></span>)</span>
 ;<br/>

<a NAME="E3:24_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">F1</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> <font color="Maroon">F2</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>thus </b><font color="Maroon">F1</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F2</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:24_1_2"/>
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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