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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>)</span>)</span> ) iff <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>  implies  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>)</span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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><br/><a NAME="E2:20_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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_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>, <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="firebrick">:: 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="firebrick">::  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/>

<a NAME="E1:20_1_1"/><span class="lab"><font color="Green" title="E18">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> <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>) <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="c3">b</font>,<font color="Maroon" title="c4">c</font>)
 
;<br/>
<a NAME="E2:20_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="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) &amp;  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="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="c2">a</font>,<font color="Maroon" title="c4">c</font>) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:20_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="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="#E1:20_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E1:20_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:20_1"/>
<font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D9" target="_self" title="METRIC_1:def.9">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:20"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>)</span>)</span></span><br/>

<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span></span><br/>

<a NAME="E3:20"/>
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="metric_1.html#D9" target="_self" title="METRIC_1:def.9">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:20"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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="ref" href="metric_1.html#D5" target="_self" title="METRIC_1:def.5">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
