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<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="metric_1.html#L1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K3" target="_self">dist_toler</a> <font color="Maroon">M</font>,<font color="Maroon">a</font>
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/><a NAME="E5:42_1"/><i><font color="Green">E31</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E6:42_1"/>
<font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="taxonom1.html#R1" target="_self">are_in_tolerance_wrt</a> <font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:42_1"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="taxonom1.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E8:42_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K1" target="_self">low_toler</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>,<font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E2:42"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="taxonom1.html#K3" target="_self">dist_toler</a> <font color="Maroon">M</font>,<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="taxonom1.html#K1" target="_self">low_toler</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:42_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K1" target="_self">low_toler</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>,<font color="Maroon">a</font>
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:42_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> <b>by </b><i><a class="txt" href="taxonom1.html#E3:42_2"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E5:42_2"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E6:42_2"/><b>then </b>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:42_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="taxonom1.html#E3:42_2"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E7:42_2"/><b>then </b>
<font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="taxonom1.html#R1" target="_self">are_in_tolerance_wrt</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="taxonom1.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E8:42_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K3" target="_self">dist_toler</a> <font color="Maroon">M</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E3:42_2"><i><font color="Green">E38</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:42"/><b>then </b>
 <a href="taxonom1.html#K1" target="_self">low_toler</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>,<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="taxonom1.html#K3" target="_self">dist_toler</a> <font color="Maroon">M</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:42"/>
 <a href="taxonom1.html#K3" target="_self">dist_toler</a> <font color="Maroon">M</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="taxonom1.html#K1" target="_self">low_toler</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>,<font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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