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<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_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#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-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#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #);<br/><b>thus </b> <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<a href="metric_1.html#K3">Empty^2-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 href="metric_1.html#K3">Empty^2-to-zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
 <a href="metric_1.html#K2">dist</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/></div><b>end;</b></div>
<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_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#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-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#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #);<br/><a NAME="E1:12_1_1_2"/><i><font color="Green">E26</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="metric_1.html#K3">Empty^2-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/><b>assume </b><a NAME="E2:12_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><b>hence </b><a NAME="E3:12_1_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_3"><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#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-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#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-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#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #);<br/><a NAME="E1:12_1_1_3"/>
(  <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#K3">Empty^2-to-zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp;  <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="metric_1.html#K3">Empty^2-to-zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> &amp;  <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="metric_1.html#K3">Empty^2-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>hence </b><a NAME="E2:12_1_1_3"/>
 <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/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_4"><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#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #);<br/><b>thus </b> <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> = 
<a href="metric_1.html#K3">Empty^2-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/>.= 
0
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:12_1_1"/>
(  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #) is <a href="metric_1.html#V6">Reflexive</a> &amp;  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #) is <a href="metric_1.html#V8">symmetric</a> &amp;  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #) is <a href="sub_metr.html#V2" target="_self">Discerning</a> &amp;  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #) is <a href="metric_1.html#V9">triangle</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/>


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