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<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:6_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#T14">NORMSP_1:14</a></i>;<br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normsp_2.html#K1" target="_self">distance_by_norm_of</a> <font color="Maroon">X</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normsp_2.html#K1" target="_self">distance_by_norm_of</a> <font color="Maroon">X</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normsp_2.html#K1" target="_self">distance_by_norm_of</a> <font color="Maroon">X</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E3:6_1"/>
<span class="p1">(<span class="default"><a href="normsp_2.html#K1" target="_self">distance_by_norm_of</a> <font color="Maroon">X</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normsp_2.html#K1" target="_self">distance_by_norm_of</a> <font color="Maroon">X</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normsp_2.html#K1" target="_self">distance_by_norm_of</a> <font color="Maroon">X</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:6"/>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="normsp_2.html#K1" target="_self">distance_by_norm_of</a> <font color="Maroon">X</font></span>)</span> #) is <a href="metric_1.html#V5">triangle</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D6">METRIC_1:def 6</a></i>;<br/>


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