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

<b>set </b><font color="Maroon" title="c2">M</font> =  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="sub_metr.html#K1" title="SUB_METR:func.1">Set_to_zero</a>  #);<br/>
<a NAME="E1:28_1_1"/>
 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="sub_metr.html#K1" title="SUB_METR:func.1">Set_to_zero</a>  #) is   <a href="sub_metr.html#NM1" title="SUB_METR:NM.1">PseudoMetricSpace</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E16">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="sub_metr.html#K1" title="SUB_METR:func.1">Set_to_zero</a>  #);<br/><a NAME="E1:28_1_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sub_metr.html#K1" title="SUB_METR:func.1">Set_to_zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><b>hence </b><a NAME="E2:28_1_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="sub_metr.html#T40" target="_self" title="SUB_METR:th.40">Th40</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="sub_metr.html#K1" title="SUB_METR:func.1">Set_to_zero</a>  #);<br/><b>set </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c3">x</font>;<br/><b>set </b><font color="Maroon" title="c7">y'</font> = <font color="Maroon" title="c4">y</font>;<br/><b>set </b><font color="Maroon" title="c8">z'</font> = <font color="Maroon" title="c5">z</font>;<br/><a NAME="E1:28_1_1_1_2"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sub_metr.html#K1" title="SUB_METR:func.1">Set_to_zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E2:28_1_1_1_2"/><i><font color="Green" title="E18">A3</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sub_metr.html#K1" title="SUB_METR:func.1">Set_to_zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E3:28_1_1_1_2"/><i><font color="Green" title="E19">A4</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sub_metr.html#K1" title="SUB_METR:func.1">Set_to_zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E4:28_1_1_1_2"/><i><font color="Green" title="E20">A5</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sub_metr.html#K1" title="SUB_METR:func.1">Set_to_zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><b>thus </b><a NAME="E5:28_1_1_1_2"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="sub_metr.html#E1:28_1_1_1_2"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="sub_metr.html#E2:28_1_1_1_2"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="sub_metr.html#T42" target="_self" title="SUB_METR:th.42">Th42</a></i>;<br/><b>thus </b><a NAME="E6:28_1_1_1_2"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="sub_metr.html#E1:28_1_1_1_2"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="sub_metr.html#E3:28_1_1_1_2"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="sub_metr.html#E4:28_1_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="sub_metr.html#T43" target="_self" title="SUB_METR:th.43">Th43</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_1_1_1"/>
 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<a href="sub_metr.html#K1" title="SUB_METR:func.1">Set_to_zero</a>  #) is   <a href="sub_metr.html#NM1" title="SUB_METR:NM.1">PseudoMetricSpace</a>
 <b>by </b><i><a class="txt" href="sub_metr.html#E1:28_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a>, <a class="ref" href="metric_1.html#T3" title="METRIC_1:th.3">METRIC_1:3</a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:28_1_1"/>
(  <a href="sub_metr.html#K2" title="SUB_METR:func.2">ZeroSpace</a>  is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> &amp;  <a href="sub_metr.html#K2" title="SUB_METR:func.2">ZeroSpace</a>  is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp;  <a href="sub_metr.html#K2" title="SUB_METR:func.2">ZeroSpace</a>  is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> )
 ;<br/>


</div>
