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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">S</font> holds <br/>( ( <font color="Olive" title="b1">S</font> is <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a> implies <font color="Olive" title="b2">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> ) &amp; ( <font color="Olive" title="b1">S</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> implies <font color="Olive" title="b2">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>  st <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">S</font> holds <br/>( ( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a> implies <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> ) &amp; ( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> implies <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> ) )</span><br/>



<b>set </b><font color="Maroon" title="c3">T</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font>;<br/>
<b>let </b><font color="Maroon" title="c4">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">S</font> implies ( ( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a> implies <font color="Maroon" title="c4">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> ) &amp; ( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> implies <font color="Maroon" title="c4">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> ) ) )</span><br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a> implies <font color="Maroon" title="c4">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> ) &amp; ( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> implies <font color="Maroon" title="c4">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> ) )</span><br/>

<b>thus </b><a NAME="E2:25"/>
( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a> implies <font color="Maroon" title="c4">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> implies <font color="Maroon" title="c4">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V6" title="COMPL_SP:attr.6">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a></span><br/>

<b>let </b><font color="Maroon" title="c5">P</font> be   <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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> or <font color="Maroon" title="c5">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<b>assume </b><a NAME="E2:25_1"/>
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:25_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="compl_sp.html#E4:25_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:25_1"/>
<font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">x</font> is <a href="compl_sp.html#V2" title="COMPL_SP:attr.2">open</a>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E1:25_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="compl_sp.html#D7" target="_self" title="COMPL_SP:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E7:25_1"/>
<font color="Maroon" title="c5">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E4:25_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:25"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a></span><br/>

<b>let </b><font color="Maroon" title="c5">P</font> be   <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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> or <font color="Maroon" title="c5">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>

<b>assume </b><a NAME="E4:25"/>
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:25"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="compl_sp.html#E6:25"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E8:25"/>
<font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">x</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E3:25"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="compl_sp.html#D8" target="_self" title="COMPL_SP:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E9:25"/>
<font color="Maroon" title="c5">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E6:25"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
