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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="4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E40</font></i>: 
<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">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:4_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">b</font> <a href="hidden.html#R1">=</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>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#T10">NORMSP_1:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:4"/>
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#V3">discerning</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D4">METRIC_1:def 4</a></i>;<br/>


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