<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>   <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>   <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</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">T</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  implies <font color="Maroon" title="c1">T</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">TM</font> =  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E1:15"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">T</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#D5" onmouseover="rs('pcomps_1/D5')" onmouseout="rh()">PCOMPS_1:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:15"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">T</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</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="c1">T</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">S2</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tbsp_1.html#D5" target="_self" onmouseover="rs('tbsp_1/D5')" onmouseout="rh()">TBSP_1:def 5</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="c3">S2</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>  implies <font color="Maroon" title="c3">S2</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:15"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c3">S2</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</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="c3">S2</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> </span><br/>

<a NAME="E4:15"/>
<font color="Maroon" title="c3">S2</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:15_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <span class="p1"> { <span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> :  ex <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> ) </span> } </span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> :  ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> ) </span> } </span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E1:15_1_1"/>
<font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> :  ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> ) </span> } </span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E2:15_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> :  ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> ) </span> } </span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <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">T</font></span>)</span>] <span class="kw">means </span> ex <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> :  ex <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> ) </span> } </span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:15_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
 for <font color="Olive" title="b1">B</font> being   <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">T</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">F</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="subset_1.html#S3" onmouseover="rs('subset_1/S3')" onmouseout="rh()">SUBSET_1:sch 3</a>();<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>] <span class="kw">means </span> ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> );<br/>
<span class="kw">set </span><font color="Maroon" title="c5">B0</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> ;<br/>
<a NAME="E4:15_1"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" onmouseover="rs('domain_1/S7')" onmouseout="rh()">DOMAIN_1:sch 7</a>();<br/></span>
<a NAME="E5:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">B0</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</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">T</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c4">F</font> as   <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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">T</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">G</font> =  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font>;<br/>
<a NAME="E8:15_1"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">B0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="pcomps_1.html#D2" onmouseover="rs('pcomps_1/D2')" onmouseout="rh()">PCOMPS_1:def 2</a></span>;<br/>
<a NAME="E9:15_1"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
 <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font> is  <a href="finset_1.html#V3" title="FINSET_1:attr.3">centered</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:15_1_2"/>
 <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1"><span class="lab"><font color="Green" title="E14">A8</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="finset_1.html#D3" onmouseover="rs('finset_1/D3')" onmouseout="rh()">FINSET_1:def 3</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">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font> or  not <font color="Olive">b<sub>1</sub></font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  or  not  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">H</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c9">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font> or  not <font color="Maroon" title="c9">H</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  or  not  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:15_1_2"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E3:15_1_2"/><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">and </span><br/><a NAME="E4:15_1_2"/><span class="lab"><font color="Green" title="E18">A12</font></span>: 
<font color="Maroon" title="c9">H</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">  not  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E5:15_1_2"/><span class="lab"><font color="Green" title="E19">A13</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</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">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_2"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E6:15_1_2"/>
<font color="Maroon" title="c9">H</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">B</font>, <font color="Maroon" title="c11">C</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D8" onmouseover="rs('ordinal1/D8')" onmouseout="rh()">ORDINAL1: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"> (  not <font color="Maroon" title="c10">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">H</font> or  not <font color="Maroon" title="c11">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">H</font> or <font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c11">C</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15_1_2_1"/><span class="lab"><font color="Green" title="E20">A14</font></span>: 
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">H</font>
 <span class="kw">and </span><br/><a NAME="E2:15_1_2_1"/><span class="lab"><font color="Green" title="E21">A15</font></span>: 
<font color="Maroon" title="c11">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">H</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"> <font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c11">C</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">B</font> = <font color="Maroon" title="c10">B</font>, <font color="Maroon" title="c13">C</font> = <font color="Maroon" title="c11">C</font> 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">T</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_2"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="txt" href="#E1:15_1_2_1"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="txt" href="#E2:15_1_2_1"><span class="lab"><font color="Green" title="E21">A15</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">V</font> being   <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">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:15_1_2_1"/><span class="lab"><font color="Green" title="E22">A16</font></span>: 
<font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c14">V</font>
 <span class="kw">and </span><br/><a NAME="E6:15_1_2_1"/><span class="lab"><font color="Green" title="E23">A17</font></span>: 
<font color="Maroon" title="c14">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_2"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="txt" href="#E1:15_1_2_1"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="ref" href="pcomps_1.html#D2" onmouseover="rs('pcomps_1/D2')" onmouseout="rh()">PCOMPS_1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:15_1_2_1"/><span class="lab"><font color="Green" title="E24">A18</font></span>: 
<font color="Maroon" title="c14">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> :  ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E6:15_1_2_1"><span class="lab"><font color="Green" title="E23">A17</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">W</font> being   <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">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:15_1_2_1"/><span class="lab"><font color="Green" title="E25">A19</font></span>: 
<font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c16">W</font>
 <span class="kw">and </span><br/><a NAME="E11:15_1_2_1"/><span class="lab"><font color="Green" title="E26">A20</font></span>: 
<font color="Maroon" title="c16">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_2"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="txt" href="#E2:15_1_2_1"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="ref" href="pcomps_1.html#D2" onmouseover="rs('pcomps_1/D2')" onmouseout="rh()">PCOMPS_1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">q</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E13:15_1_2_1"/><span class="lab"><font color="Green" title="E27">A21</font></span>: 
<font color="Maroon" title="c16">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> :  ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c17">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E11:15_1_2_1"><span class="lab"><font color="Green" title="E26">A20</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1_2_1_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"> ( ( <font color="Maroon" title="c17">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Maroon" title="c14">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">W</font> ) or ( <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">q</font> &amp; <font color="Maroon" title="c16">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">V</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2_1_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="E1:15_1_2_1_1_1"/>
( <font color="Maroon" title="c17">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">p</font> or <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:15_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E28">A22</font></span>: 
<font color="Maroon" title="c17">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">p</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"> <font color="Maroon" title="c14">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">W</font></span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:15_1_2_1_1_1_1"/>
<font color="Maroon" title="c14">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">W</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1_2_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</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"> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">W</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E3:15_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A23</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font>
 <span class="kw">and </span><br/><a NAME="E4:15_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A24</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_2_1"><span class="lab"><font color="Green" title="E24">A18</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:15_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A25</font></span>: 
<font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">and </span><br/><a NAME="E7:15_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A26</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A24</font></span></a></span>;<br/>
<a NAME="E8:15_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c17">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E28">A22</font></span></a>, <a class="txt" href="#E6:15_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:15_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15_1_2_1"><span class="lab"><font color="Green" title="E27">A21</font></span></a>, <a class="txt" href="#E3:15_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E29">A23</font></span></a>, <a class="txt" href="#E7:15_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A26</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:15_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E28">A27</font></span>: 
<font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">q</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"> <font color="Maroon" title="c16">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">V</font></span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:15_1_2_1_1_1_2"/>
<font color="Maroon" title="c16">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">V</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1_2_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">W</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">W</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"> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E3:15_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E29">A28</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font>
 <span class="kw">and </span><br/><a NAME="E4:15_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A29</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c17">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15_1_2_1"><span class="lab"><font color="Green" title="E27">A21</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:15_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A30</font></span>: 
<font color="Maroon" title="c17">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">and </span><br/><a NAME="E7:15_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E32">A31</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E30">A29</font></span></a></span>;<br/>
<a NAME="E8:15_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E28">A27</font></span></a>, <a class="txt" href="#E6:15_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:15_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_2_1"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="txt" href="#E3:15_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="txt" href="#E7:15_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E32">A31</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E15:15_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C</font> or <font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_2_1"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="txt" href="#E10:15_1_2_1"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="ref" href="pre_topc.html#T19" onmouseover="rs('pre_topc/T19')" onmouseout="rh()">PRE_TOPC:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:15_1_2_1"/>
<font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c11">C</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</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>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">m</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E8:15_1_2"/><span class="lab"><font color="Green" title="E20">A32</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">H</font>
 <span class="kw">and </span><br/><a NAME="E9:15_1_2"/><span class="lab"><font color="Green" title="E21">A33</font></span>: 
 for <font color="Olive" title="b1">C</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">H</font> holds <br/><font color="Maroon" title="c10">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="txt" href="#E4:15_1_2"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="ref" href="finset_1.html#T11" onmouseover="rs('finset_1/T11')" onmouseout="rh()">FINSET_1:11</a></span>;<br/>
<a NAME="E10:15_1_2"/><span class="lab"><font color="Green" title="E22">A34</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c9">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="txt" href="#E9:15_1_2"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="ref" href="setfam_1.html#T5" onmouseover="rs('setfam_1/T5')" onmouseout="rh()">SETFAM_1:5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">m</font> = <font color="Maroon" title="c10">m</font> 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">T</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_2"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="txt" href="#E8:15_1_2"><span class="lab"><font color="Green" title="E20">A32</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">A</font> being   <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">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E13:15_1_2"/><span class="lab"><font color="Green" title="E23">A35</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c12">A</font>
 <span class="kw">and </span><br/><a NAME="E14:15_1_2"/><span class="lab"><font color="Green" title="E24">A36</font></span>: 
<font color="Maroon" title="c12">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_2"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="txt" href="#E8:15_1_2"><span class="lab"><font color="Green" title="E20">A32</font></span></a>, <a class="ref" href="pcomps_1.html#D2" onmouseover="rs('pcomps_1/D2')" onmouseout="rh()">PCOMPS_1:def 2</a></span>;<br/>
<a NAME="E15:15_1_2"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E14:15_1_2"><span class="lab"><font color="Green" title="E24">A36</font></span></a></span>;<br/>
<a NAME="E16:15_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15_1_2"><span class="lab"><font color="Green" title="E23">A35</font></span></a>, <a class="ref" href="pre_topc.html#T18" onmouseover="rs('pre_topc/T18')" onmouseout="rh()">PRE_TOPC:18</a>, <a class="ref" href="xboole_1.html#T3" onmouseover="rs('xboole_1/T3')" onmouseout="rh()">XBOOLE_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:15_1_2"/>
 not  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15_1_2"><span class="lab"><font color="Green" title="E22">A34</font></span></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>
<a NAME="E10:15_1"/>
 <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font> is  <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#T11" onmouseover="rs('pcomps_1/T11')" onmouseout="rh()">PCOMPS_1:11</a></span>;<br/>
<a NAME="E11:15_1"/><span class="kw">then </span>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E9:15_1"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="compts_1.html#T4" onmouseover="rs('compts_1/T4')" onmouseout="rh()">COMPTS_1:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">c</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E13:15_1"/><span class="lab"><font color="Green" title="E16">A37</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T4" onmouseover="rs('subset_1/T4')" onmouseout="rh()">SUBSET_1:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">c</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tbsp_1.html#D2" target="_self" onmouseover="rs('tbsp_1/D2')" onmouseout="rh()">TBSP_1:def 2</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">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">m</font></span>)</span>,<font color="Maroon" title="c10">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="c11">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">m</font></span>)</span>,<font color="Maroon" title="c10">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E15:15_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</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">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">m</font></span>)</span>,<font color="Maroon" title="c10">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font></span><br/>

<a NAME="E16:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A38</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T215" onmouseover="rs('xreal_1/T215')" onmouseout="rh()">XREAL_1:215</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E18:15_1"/><span class="lab"><font color="Green" title="E18">A39</font></span>: 
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/>
<a NAME="E19:15_1"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c10">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15_1"><span class="lab"><font color="Green" title="E17">A38</font></span></a>, <a class="ref" href="metric_1.html#T1" onmouseover="rs('metric_1/T1')" onmouseout="rh()">METRIC_1:1</a></span>;<br/>
<a NAME="E20:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A40</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c10">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">Z</font> =  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c10">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</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">T</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/>
<a NAME="E22:15_1"/>
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c10">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#T29" onmouseover="rs('pcomps_1/T29')" onmouseout="rh()">PCOMPS_1:29</a></span>;<br/>
<a NAME="E23:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A41</font></span>: 
<font color="Maroon" title="c13">Z</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> );<br/>
<span class="kw">set </span><font color="Maroon" title="c14">B</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> ;<br/>
<a NAME="E24:15_1"/><span class="lab"><font color="Green" title="E21">A42</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" onmouseover="rs('domain_1/S7')" onmouseout="rh()">DOMAIN_1:sch 7</a>();<br/></span>
<a NAME="E25:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A43</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">B</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</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">T</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E24:15_1"><span class="lab"><font color="Green" title="E21">A42</font></span></a></span>;<br/>
<a NAME="E27:15_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c15">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">clf</a> <font color="Maroon" title="c7">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:15_1"><span class="lab"><font color="Green" title="E22">A43</font></span></a>, <a class="ref" href="pcomps_1.html#D2" onmouseover="rs('pcomps_1/D2')" onmouseout="rh()">PCOMPS_1:def 2</a></span>;<br/>
<a NAME="E28:15_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c15">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15_1"><span class="lab"><font color="Green" title="E16">A37</font></span></a>, <a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a></span>;<br/>
<a NAME="E29:15_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c13">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:15_1"><span class="lab"><font color="Green" title="E19">A40</font></span></a>, <a class="txt" href="#E23:15_1"><span class="lab"><font color="Green" title="E20">A41</font></span></a>, <a class="ref" href="pre_topc.html#D7" onmouseover="rs('pre_topc/D7')" onmouseout="rh()">PRE_TOPC:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E31:15_1"/><span class="lab"><font color="Green" title="E23">A44</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" onmouseover="rs('xboole_0/T4')" onmouseout="rh()">XBOOLE_0:4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">p</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">  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">m</font></span>)</span>,<font color="Maroon" title="c10">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c17">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font> implies  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">m</font></span>)</span>,<font color="Maroon" title="c10">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font> )</span><br/>

<a NAME="E32:15_1"/><span class="lab"><font color="Green" title="E24">A45</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:15_1"><span class="lab"><font color="Green" title="E23">A44</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E33:15_1"/><span class="lab"><font color="Green" title="E25">A46</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:15_1"><span class="lab"><font color="Green" title="E23">A44</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">g</font> = <font color="Maroon" title="c16">g</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E36:15_1"/><span class="lab"><font color="Green" title="E26">A47</font></span>: 
<font color="Maroon" title="c18">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font>
 <span class="kw">and </span><br/><a NAME="E37:15_1"/><span class="lab"><font color="Green" title="E27">A48</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:15_1"><span class="lab"><font color="Green" title="E24">A45</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E39:15_1"/><span class="lab"><font color="Green" title="E28">A49</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">and </span><br/><a NAME="E40:15_1"/><span class="lab"><font color="Green" title="E29">A50</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:15_1"><span class="lab"><font color="Green" title="E27">A48</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E41:15_1"/>
<font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</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">  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">m</font></span>)</span>,<font color="Maroon" title="c10">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font></span><br/>

<a NAME="E42:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A51</font></span>: 
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:15_1"><span class="lab"><font color="Green" title="E18">A39</font></span></a>, <a class="txt" href="#E39:15_1"><span class="lab"><font color="Green" title="E28">A49</font></span></a></span>;<br/>
<a NAME="E43:15_1"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c10">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:15_1"><span class="lab"><font color="Green" title="E25">A46</font></span></a>, <a class="txt" href="#E36:15_1"><span class="lab"><font color="Green" title="E26">A47</font></span></a>, <a class="txt" href="#E40:15_1"><span class="lab"><font color="Green" title="E29">A50</font></span></a>, <a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</a></span>;<br/>
<a NAME="E44:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A52</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c10">c</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:15_1"><span class="lab"><font color="Green" title="E30">A51</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<a NAME="E45:15_1"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">m</font></span>)</span>,<font color="Maroon" title="c10">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#K3" title="METRIC_1:func.3">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c10">c</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" onmouseover="rs('metric_1/T4')" onmouseout="rh()">METRIC_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E46:15_1"/>
 <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">m</font></span>)</span>,<font color="Maroon" title="c10">c</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:15_1"><span class="lab"><font color="Green" title="E31">A52</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="E5:15"/>
<font color="Maroon" title="c3">S2</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</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>
