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<b>let </b><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</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"><span class="p2"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>,<span class="p2"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span></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="E1:28_1_2"/><i><font color="Green" title="E22">A3</font></i>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">y2</font>, <font color="Olive" title="b5">x3</font>, <font color="Olive" title="b6">y3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b7">x</font>, <font color="Olive" title="b8">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>  st <font color="Olive" title="b7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b5">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Olive" title="b8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b2">y1</font>,<font color="Olive" title="b4">y2</font>,<font color="Olive" title="b6">y3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> holds <br/><font color="Maroon" title="c1">f1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b7">x</font>,<font color="Olive" title="b8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b3">x2</font>,<font color="Olive" title="b4">y2</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b5">x3</font>,<font color="Olive" title="b6">y3</font></span>)</span>
 <b>and </b><br/><a NAME="E2:28_1_2"/><i><font color="Green" title="E23">A4</font></i>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">y2</font>, <font color="Olive" title="b5">x3</font>, <font color="Olive" title="b6">y3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b7">x</font>, <font color="Olive" title="b8">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>  st <font color="Olive" title="b7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b5">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Olive" title="b8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b2">y1</font>,<font color="Olive" title="b4">y2</font>,<font color="Olive" title="b6">y3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> holds <br/><font color="Maroon" title="c2">f2</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b7">x</font>,<font color="Olive" title="b8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b3">x2</font>,<font color="Olive" title="b4">y2</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b5">x3</font>,<font color="Olive" title="b6">y3</font></span>)</span>
 ;<br/>

<a NAME="E3:28_1_2"/>
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 <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds  <font color="Maroon" title="c1">f1</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> <font color="Maroon" title="c2">f2</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>;<br/>

<b>consider </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">x3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:28_1_2_1"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T15" title="DOMAIN_1:th.15">DOMAIN_1:15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c9">y2</font>, <font color="Maroon" title="c10">y3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:28_1_2_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">y2</font>,<font color="Maroon" title="c10">y3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T15" title="DOMAIN_1:th.15">DOMAIN_1:15</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c1">f1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c8">y1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c9">y2</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">x3</font>,<font color="Maroon" title="c10">y3</font></span>)</span>
<b>by </b><i><a class="txt" href="metric_4.html#E1:28_1_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="metric_4.html#E2:28_1_2_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="metric_4.html#E4:28_1_2_1"><i><font color="Green" title="E25">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">f2</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="txt" href="metric_4.html#E2:28_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="metric_4.html#E2:28_1_2_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="metric_4.html#E4:28_1_2_1"><i><font color="Green" title="E25">A6</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:28_1_2"/>
<font color="Maroon" title="c1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


</div>
