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<b>let </b><font color="Maroon">M</font> be   <a href="sub_metr.html#NM1">PseudoMetricSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">x</font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">y</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 ;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:14_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : <font color="Maroon">x</font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Olive">q</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E5:14_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x</font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Olive">q</font> )
 ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/><a NAME="E7:14_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : <font color="Maroon">y</font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Olive">s</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E8:14_1"/><b>then </b>
 ex <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Olive">s</font> )
 ;<br/><b>hence </b><a NAME="E9:14_1"/>
<font color="Maroon">x</font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:14"/>
<font color="Maroon">x</font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:14"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 
;<br/>
<a NAME="E4:14"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:14"/>
<font color="Maroon">x</font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">y</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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