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<div class="add">

<b>reconsider </b><font color="Maroon" title="c1">M</font> =  <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>  #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#V1" title="METRIC_1:attr.1">strict</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c1">M</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V1" title="METRIC_1:attr.1">strict</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> &amp; not <font color="Maroon" title="c1">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )</span><br/>

<a NAME="E2:15_1_1"/>
( <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:15_1_1_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D7" target="_self" title="METRIC_1:def.7">METRIC_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D3" target="_self" title="METRIC_1:def.3">METRIC_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<b>thus </b><a NAME="E1:15_1_1_1_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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="txt" href="metric_1.html#E3"><i><font color="Green" title="E2">Lm3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:15_1_1_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D8" target="_self" title="METRIC_1:def.8">METRIC_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D4" target="_self" title="METRIC_1:def.4">METRIC_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )</span><br/>

<b>assume </b><a NAME="E1:15_1_1_1_2"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font></span><br/>

<b>hence </b><a NAME="E2:15_1_1_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="metric_1.html#E3"><i><font color="Green" title="E2">Lm3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:15_1_1_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D9" target="_self" title="METRIC_1:def.9">METRIC_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D5" target="_self" title="METRIC_1:def.5">METRIC_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><br/>

<b>thus </b><a NAME="E1:15_1_1_1_3"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="metric_1.html#E4"><i><font color="Green" title="E3">Lm4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:15_1_1_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D10" target="_self" title="METRIC_1:def.10">METRIC_1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D6" target="_self" title="METRIC_1:def.6">METRIC_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span>)</span></span><br/>

<b>thus </b><a NAME="E1:15_1_1_1_4"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_1.html#E5"><i><font color="Green" title="E4">Lm5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_1_1"/>
( <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V1" title="METRIC_1:attr.1">strict</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> &amp; not <font color="Maroon" title="c1">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
