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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:9_3_1_1"/>
( <font color="Maroon" title="c1">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <font color="Maroon" title="c1">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_3_1_1_1"/>
<font color="Maroon" title="c1">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> </span><br/><div class="add"><a NAME="E2:9_3_1_1_1"/><span class="kw">then </span>
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ;<br/><span class="kw">hence </span><a NAME="E3:9_3_1_1_1"/>
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_3_1_1_2"/>
 not <font color="Maroon" title="c1">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">m</font> = <font color="Maroon" title="c1">M</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c3">TM</font> =  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">CM</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">M</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">m</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">CTM</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">TMM</font> =  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c4">CM</font></span>)</span>;<br/><a NAME="E5:9_3_1_1_2"/><span class="lab"><font color="Green" title="E6">A8</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c4">CM</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CM</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#D2" onmouseover="rs('topmetr/D2')" onmouseout="rh()">TOPMETR:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">D</font> =  the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c4">CM</font></span>)</span> 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 <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></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/><span class="kw">take </span>
<font color="Maroon" title="c7">D</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pcomps_1.html#D8" onmouseover="rs('pcomps_1/D8')" onmouseout="rh()">PCOMPS_1:def 8</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"> ( <font color="Maroon" title="c7">D</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 <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp;  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<font color="Maroon" title="c7">D</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 <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> )</span><br/><a NAME="E7:9_3_1_1_2"/><span class="lab"><font color="Green" title="E7">A9</font></span>: 
<font color="Maroon" title="c7">D</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 <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_3_1_1_2"><span class="lab"><font color="Green" title="E6">A8</font></span></a>, <a class="ref" href="pcomps_1.html#T35" onmouseover="rs('pcomps_1/T35')" onmouseout="rh()">PCOMPS_1:35</a></span>;<br/><a NAME="E8:9_3_1_1_2"/>
( <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">CTM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c4">CM</font></span>)</span> &amp;  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hausdorf.html#T16" onmouseover="rs('hausdorf/T16')" onmouseout="rh()">HAUSDORF:16</a>, <a class="ref" href="tsep_1.html#T2" onmouseover="rs('tsep_1/T2')" onmouseout="rh()">TSEP_1:2</a></span>;<br/><span class="kw">then </span> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> = 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c4">CM</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_3_1_1_2"><span class="lab"><font color="Green" title="E6">A8</font></span></a>, <a class="ref" href="tsep_1.html#T5" onmouseover="rs('tsep_1/T5')" onmouseout="rh()">TSEP_1:5</a></span>
<br/>.= 
 <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<font color="Maroon" title="c7">D</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_3_1_1_2"><span class="lab"><font color="Green" title="E6">A8</font></span></a>, <a class="txt" href="#E7:9_3_1_1_2"><span class="lab"><font color="Green" title="E7">A9</font></span></a>, <a class="ref" href="pcomps_1.html#D7" onmouseover="rs('pcomps_1/D7')" onmouseout="rh()">PCOMPS_1:def 7</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:9_3_1_1_2"/>
( <font color="Maroon" title="c7">D</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 <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp;  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>,<font color="Maroon" title="c7">D</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 <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_3_1_1_2"><span class="lab"><font color="Green" title="E6">A8</font></span></a>, <a class="ref" href="pcomps_1.html#T35" onmouseover="rs('pcomps_1/T35')" onmouseout="rh()">PCOMPS_1:35</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></div><span class="kw">end;</span></div>

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