<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">M</font> =  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #);<br/>
<a NAME="E1:29_1_1"/>
 <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #) is   <a href="sub_metr.html#NM1">PseudoMetricSpace</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #);<br/><a NAME="E1:29_1_1_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><b>hence </b><a NAME="E2:29_1_1_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #), <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #);<br/><b>set </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>;<br/><b>set </b><font color="Maroon">y'</font> = <font color="Maroon">y</font>;<br/><b>set </b><font color="Maroon">z'</font> = <font color="Maroon">z</font>;<br/><a NAME="E1:29_1_1_1_2"/><i><font color="Green">E25</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E2:29_1_1_1_2"/><i><font color="Green">E26</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E3:29_1_1_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E4:29_1_1_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><b>thus </b><a NAME="E5:29_1_1_1_2"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E6:29_1_1_1_2"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="sub_metr.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1_1"/>
 <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #) is   <a href="sub_metr.html#NM1">PseudoMetricSpace</a>
 <b>by </b><i>, <a class="ref" href="metric_1.html#T1">METRIC_1:1</a>, <a class="ref" href="metric_1.html#T3">METRIC_1:3</a>, <a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_1_1"/>
(  <a href="sub_metr.html#K2" target="_self">ZeroSpace</a>  is <a href="metric_1.html#V6">Reflexive</a> &amp;  <a href="sub_metr.html#K2" target="_self">ZeroSpace</a>  is <a href="metric_1.html#V8">symmetric</a> &amp;  <a href="sub_metr.html#K2" target="_self">ZeroSpace</a>  is <a href="metric_1.html#V9">triangle</a> )
 ;<br/>


</div>
