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<b>consider </b><font color="Maroon" title="c1">MS</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="metric_1.html#V1" title="METRIC_1:attr.1">strict</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/>
<b>take </b><font color="Maroon" title="c2">TS</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">MS</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c3">f</font> = the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">MS</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TS</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>thus </b><a NAME="E3:48_1_1"/>
<font color="Maroon" title="c2">TS</font> is <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>
 ;<br/>

<b>take </b>
<font color="Maroon" title="c3">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pcomps_1.html#D9" target="_self" title="PCOMPS_1:def.9">PCOMPS_1:def 9</a><br/>

<b>thus </b><a NAME="E4:48_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TS</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T39" target="_self" title="PCOMPS_1:th.39">Th39</a></i>;<br/>

<b>hence </b><a NAME="E5:48_1_1"/>
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">TS</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">TS</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D8" target="_self" title="PCOMPS_1:def.8">Def8</a></i>;<br/>


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