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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">V</font>, <font color="Olive" title="b2">Q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> <br/> for <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</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>   st <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Olive" title="b3">v1</font> &amp; <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Olive" title="b4">v2</font> holds <br/><font color="Olive" title="b3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v2</font></span><br/><b>let </b><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">Q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</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">v1</font>, <font color="Olive" title="b2">v2</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>   st <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Olive" title="b1">v1</font> &amp; <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Olive" title="b2">v2</font> holds <br/><font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</font></span><br/><b>let </b><font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Maroon" title="c5">v2</font> implies <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> )</span><br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Maroon" title="c5">v2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">p1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E3:48"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</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#T23" target="_self" title="METRIC_2:th.23">Th23</a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="#E3:48"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="metric_2.html#T9" target="_self" title="METRIC_2:th.9">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E6:48"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q1</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#T23" target="_self" title="METRIC_2:th.23">Th23</a></i>;<br/>
<a NAME="E7:48"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c7">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 
<b>by </b><i><a class="txt" href="#E6:48"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="metric_2.html#T9" target="_self" title="METRIC_2:th.9">Th9</a></i>;<br/>
<a NAME="E8:48"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font>
 
<b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E4:48"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="metric_2.html#D5" target="_self" title="METRIC_2:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E9:48"/>
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font>
 <b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E4:48"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="#E7:48"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="metric_2.html#D5" target="_self" title="METRIC_2:def.5">Def5</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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