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<div><i><font color="Green" title="E11">A1</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" 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 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  #);<br/><b>thus </b> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> = 
<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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 href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="sub_metr.html#T6" target="_self" title="SUB_METR:th.6">Th6</a></i>
<br/>.= 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">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/></div><b>end;</b></div>
<div><i><font color="Green" title="E12">A2</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" 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 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  #);<br/><a NAME="E1:12_1_1_2"/><i><font color="Green" title="E13">A3</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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/><b>assume </b><a NAME="E2:12_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 ;<br/><b>hence </b><a NAME="E3:12_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="sub_metr.html#E1:12_1_1_2"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="sub_metr.html#T5" target="_self" title="SUB_METR:th.5">Th5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E13">A4</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" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">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>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  #);<br/><a NAME="E1:12_1_1_3"/>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">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>hence </b><a NAME="E2:12_1_1_3"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">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="c2">x</font>,<font color="Maroon" title="c3">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="c3">y</font>,<font color="Maroon" title="c4">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="sub_metr.html#T7" target="_self" title="SUB_METR:th.7">Th7</a></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" title="c2">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>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  #);<br/><b>thus </b> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font> = 
<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">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 href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="sub_metr.html#T4" target="_self" title="SUB_METR:th.4">Th4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:12_1_1"/>
(  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  #) is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> &amp;  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  #) is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp;  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  #) is <a href="sub_metr.html#V2" title="SUB_METR:attr.2">Discerning</a> &amp;  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a>  #) is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> )
 <b>by </b><i><a class="txt" href="sub_metr.html#E1:12_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="sub_metr.html#E2:12_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="sub_metr.html#E3:12_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="sub_metr.html#T14" target="_self" title="SUB_METR:th.14">Th14</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/>


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