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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c1">X</font></span>)</span> #) is  <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>)</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">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</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> <span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>)</span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) = 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D1" target="_self" onmouseover="rs('normsp_2/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T7" onmouseover="rs('normsp_1/T7')" onmouseout="rh()">NORMSP_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</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="ref" href="normsp_2.html#D1" target="_self" onmouseover="rs('normsp_2/D1')" onmouseout="rh()">Def1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:5"/>
 the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c1">X</font></span>)</span> #) is  <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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