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<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">M</font>,the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> #);<br/><b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c6">b'</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c7">c'</font> = <font color="Maroon" title="c4">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> ;<br/><i><font color="Green" title="E6">A1</font></i>:  <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> = 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:11_1"/>
(  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></i>;<br/> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> = 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c5">a'</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
;<br/><b>hence </b><a NAME="E5:11_1"/>
 <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> <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="c2">a</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E2:11_1"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/><i><font color="Green" title="E7">A2</font></i>:  <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> = 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
;<br/> <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> = 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c7">c'</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
;<br/><b>hence </b><a NAME="E8:11_1"/>
 <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>
 <b>by </b><i><a class="txt" href="topmetr.html#E2:11_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="topmetr.html#E6:11_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:11"/>
 <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">M</font>,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#NM1" title="METRIC_1:NM.1">MetrSpace</a>
 <b>by </b><i><a class="ref" href="metric_1.html#T6" title="METRIC_1:th.6">METRIC_1:6</a></i>;<br/>


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