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<span class="kw">reconsider </span><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>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c1">M</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><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> <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">hence </span><a NAME="E2:15_1_1_1"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="metric_1.html#E3"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:15_1_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E1:15_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> 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 class="kw">by</span> <span class="lab"><a class="txt" href="metric_1.html#E4"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:15_1_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E1:15_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="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 class="kw">by</span> <span class="lab"><a class="txt" href="metric_1.html#E5"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:15_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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E1:15_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="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 class="kw">by</span> <span class="lab"><a class="txt" href="metric_1.html#E3"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:15_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E4:15_1_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="metric_1.html#D7" target="_self" title="METRIC_1:def.7">Def7</a>, <a class="ref" href="metric_1.html#D8" target="_self" title="METRIC_1:def.8">Def8</a>, <a class="ref" href="metric_1.html#D9" target="_self" title="METRIC_1:def.9">Def9</a>, <a class="ref" href="metric_1.html#D10" target="_self" title="METRIC_1:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
