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<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="sub_metr.html#NM1" title="SUB_METR:NM.1">PseudoMetricSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  iff <font color="Olive" title="b1">x</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Olive" title="b2">y</font> )</span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  iff <font color="Maroon" title="c2">x</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c3">y</font> )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  )</span>
<div class="add"><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c3">y</font></span><br/><b>then consider </b><font color="Maroon" title="c4">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:14_1"/><i><font color="Green" title="E10">A2</font></i>: 
 ex <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c2">x</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Olive" title="b1">q</font> )
 <b>by </b><i><a class="txt" href="#E3:14_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="#E3:14_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><a NAME="E6:14_1"/>
 ex <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c3">y</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Olive" title="b1">s</font> )
 <b>by </b><i><a class="txt" href="#E3:14_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E7:14_1"/>
<font color="Maroon" title="c2">x</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="#E4:14_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="metric_2.html#T6" target="_self" title="METRIC_2:th.6">Th6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:14"/>
<font color="Maroon" title="c2">x</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> </span><br/>

<a NAME="E3:14"/><b>then </b><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 
;<br/>
<a NAME="E4:14"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 
<b>by </b><i><a class="ref" href="metric_2.html#T9" target="_self" title="METRIC_2:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E5:14"/>
<font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 <b>by </b><i><a class="txt" href="#E3:14"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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