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

<span class="kw">let </span><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#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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">M</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  implies for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  st  ex <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b2">X</font> being  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> (  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">X</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  &amp;  ex <font color="Olive" title="b3">S</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>   <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a>  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">X</font></span>)</span> st <br/>( <font color="Olive" title="b3">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b3">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:72"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c1">M</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  st  ex <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b2">X</font> being  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> (  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">X</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  &amp;  ex <font color="Olive" title="b3">S</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>   <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a>  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">X</font></span>)</span> st <br/>( <font color="Olive" title="b3">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  &amp; <font color="Olive" title="b3">S</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b3">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">X</font> being  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> (  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">X</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  &amp;  ex <font color="Olive" title="b2">S</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>   <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a>  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">X</font></span>)</span> st <br/>( <font color="Olive" title="b2">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b2">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:72"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> (  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">X</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  &amp;  ex <font color="Olive" title="b2">S</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>   <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a>  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">X</font></span>)</span> st <br/>( <font color="Olive" title="b2">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  &amp; <font color="Olive" title="b2">S</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b2">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E4:72"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  &amp;  ex <font color="Olive" title="b1">S</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>   <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a>  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b1">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">W</font> =  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font>;<br/>
<span class="kw">thus </span><a NAME="E5:72"/>
 <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="compl_sp.html#T42" target="_self" title="COMPL_SP:th.42">Th42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>   <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a>  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">S</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b1">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">TW</font> =  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c4">X</font><span class="kw"> such that </span><br/><a NAME="E7:72"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dickson.html#T3" title="DICKSON:th.3">DICKSON:3</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E8:72"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/>
<a NAME="E9:72"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<a NAME="E2:72_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:72_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">take </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>] )</span><br/>

<span class="kw">thus </span><a NAME="E4:72_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E3:72_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">s</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:72"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E9:72"><span class="lab"><font color="Green" title="E59">A5</font></span></a>);<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E12:72"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:72_2"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:72_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span><br/>

<a NAME="E3:72_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_2"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E4:72_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">x</font>) is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T55" title="SUBSET_1:th.55">SUBSET_1:55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:72_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c9">S</font> being   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:72"/><span class="lab"><font color="Green" title="E62">A10</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E12:72"><span class="lab"><font color="Green" title="E61">A8</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E63">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> implies <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:72_3"/><span class="lab"><font color="Green" title="E64">A12</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E2:72_3"/><span class="lab"><font color="Green" title="E65">A13</font></span>: 
<font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E3:72_3"/><span class="lab"><font color="Green" title="E66">A14</font></span>: 
<font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font></span><br/><a NAME="E4:72_3"/><span class="lab"><font color="Green" title="E67">A15</font></span>: 
<font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:72"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E2:72_3"><span class="lab"><font color="Green" title="E65">A13</font></span></a></span>;<br/><a NAME="E5:72_3"/><span class="lab"><font color="Green" title="E68">A16</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:72"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E1:72_3"><span class="lab"><font color="Green" title="E64">A12</font></span></a></span>;<br/><a NAME="E6:72_3"/><span class="lab"><font color="Green" title="E69">A17</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E7:72_3"/><span class="lab"><font color="Green" title="E70">A18</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:72"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E2:72_3"><span class="lab"><font color="Green" title="E65">A13</font></span></a></span>;<br/><a NAME="E8:72_3"/>
<font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:72"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E1:72_3"><span class="lab"><font color="Green" title="E64">A12</font></span></a></span>;<br/><a NAME="E9:72_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_3"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="txt" href="#E4:72_3"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="txt" href="#E6:72_3"><span class="lab"><font color="Green" title="E69">A17</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E10:72_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72_3"><span class="lab"><font color="Green" title="E68">A16</font></span></a>, <a class="txt" href="#E7:72_3"><span class="lab"><font color="Green" title="E70">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E11:72_3"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="#E1:72_3"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="txt" href="#E2:72_3"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c10">rngs</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">s</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">F</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> </span>}</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">F</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> </span>}</span>  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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T16" title="RELSET_2:th.16">RELSET_2:16</a></span>;<br/>
<a NAME="E18:72"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></span>;<br/>
<a NAME="E19:72"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A19</font></span>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></span>;<br/>
<a NAME="E20:72"/><span class="lab"><font color="Green" title="E65">A20</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">S</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_4"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:72_4"/><span class="lab"><font color="Green" title="E66">A21</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">S</font>
 <span class="kw">and </span><br/><a NAME="E4:72_4"/><span class="lab"><font color="Green" title="E67">A22</font></span>: 
<font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:72_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:72_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A23</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_4"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E7:72_4"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:72"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E3:72_4"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="txt" href="#E4:72_4"><span class="lab"><font color="Green" title="E67">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:72_4"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:72_4"><span class="lab"><font color="Green" title="E68">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">S</font> implies  not <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:72_5"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/><a NAME="E2:72_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:72"><span class="lab"><font color="Green" title="E62">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:72_5"/>
not <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E22:72"/><span class="kw">then </span>
<font color="Maroon" title="c9">S</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">R</font> being   <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>   <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a>  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E24:72"/><span class="lab"><font color="Green" title="E66">A24</font></span>: 
<font color="Maroon" title="c13">R</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> 
 <span class="kw">and </span><br/><a NAME="E25:72"/><span class="lab"><font color="Green" title="E67">A25</font></span>: 
( <font color="Maroon" title="c12">F</font> is  <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>  &amp; <font color="Maroon" title="c9">S</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c13">R</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">and </span><br/><a NAME="E26:72"/><span class="lab"><font color="Green" title="E68">A26</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b2">Si</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c13">R</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">Si</font></span>)</span> &amp; <font color="Olive" title="b2">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b3">j</font></span>)</span> where <font color="Olive" title="b3">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> </span>}</span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:72"><span class="lab"><font color="Green" title="E65">A20</font></span></a>, <a class="ref" href="compl_sp.html#T23" target="_self" title="COMPL_SP:th.23">Th23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">R9</font> = <font color="Maroon" title="c13">R</font> as   <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span> ;<br/>
<div><span class="lab"><font color="Green" title="E69">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">y</font> implies  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:72_6"/><span class="lab"><font color="Green" title="E70">A28</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font>
 <span class="kw">and </span><br/><a NAME="E2:72_6"/><span class="lab"><font color="Green" title="E71">A29</font></span>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font>
 <span class="kw">and </span><br/><a NAME="E3:72_6"/><span class="lab"><font color="Green" title="E72">A30</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c17">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:72_6"/><span class="lab"><font color="Green" title="E73">A31</font></span>: 
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">and </span><br/><a NAME="E6:72_6"/><span class="lab"><font color="Green" title="E74">A32</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_6"><span class="lab"><font color="Green" title="E71">A29</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E7:72_6"/><span class="lab"><font color="Green" title="E75">A33</font></span>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y1</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:72"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E5:72_6"><span class="lab"><font color="Green" title="E73">A31</font></span></a>, <a class="txt" href="#E6:72_6"><span class="lab"><font color="Green" title="E74">A32</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:72_6"/><span class="lab"><font color="Green" title="E76">A34</font></span>: 
<font color="Maroon" title="c18">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">and </span><br/><a NAME="E10:72_6"/><span class="lab"><font color="Green" title="E77">A35</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_6"><span class="lab"><font color="Green" title="E70">A28</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E11:72_6"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x1</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:72"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E9:72_6"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E10:72_6"><span class="lab"><font color="Green" title="E77">A35</font></span></a></span>;<br/><a NAME="E12:72_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_6"><span class="lab"><font color="Green" title="E72">A30</font></span></a>, <a class="txt" href="#E7:72_6"><span class="lab"><font color="Green" title="E75">A33</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E13:72_6"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">i</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">R9</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c15">i</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c16">Si</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> <span class="p2">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:72_7"/><span class="lab"><font color="Green" title="E70">A36</font></span>: 
<font color="Maroon" title="c13">R</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">Si</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:72_7"/><span class="lab"><font color="Green" title="E71">A37</font></span>: 
<font color="Maroon" title="c16">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:72"><span class="lab"><font color="Green" title="E68">A26</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">SI</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">Si</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c19">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">SI</font> &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">SI</font> implies  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c19">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:72_7_1"/><span class="lab"><font color="Green" title="E72">A38</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">SI</font>
 <span class="kw">and </span><br/><a NAME="E2:72_7_1"/><span class="lab"><font color="Green" title="E73">A39</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">SI</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c19">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c20">xS</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:72_7_1"/><span class="lab"><font color="Green" title="E74">A40</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">xS</font>
 <span class="kw">and </span><br/><a NAME="E5:72_7_1"/><span class="lab"><font color="Green" title="E75">A41</font></span>: 
<font color="Maroon" title="c20">xS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Si</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_7_1"><span class="lab"><font color="Green" title="E72">A38</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">j1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:72_7_1"/><span class="lab"><font color="Green" title="E76">A42</font></span>: 
<font color="Maroon" title="c20">xS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c21">j1</font>
 <span class="kw">and </span><br/><a NAME="E8:72_7_1"/>
<font color="Maroon" title="c21">j1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_7"><span class="lab"><font color="Green" title="E71">A37</font></span></a>, <a class="txt" href="#E5:72_7_1"><span class="lab"><font color="Green" title="E75">A41</font></span></a></span>;<br/><a NAME="E9:72_7_1"/><span class="lab"><font color="Green" title="E77">A43</font></span>: 
<font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c21">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c21">j1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:72"><span class="lab"><font color="Green" title="E62">A10</font></span></a></span>;<br/><a NAME="E10:72_7_1"/><span class="lab"><font color="Green" title="E78">A44</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E11:72_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c21">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E12:72_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A45</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_7_1"><span class="lab"><font color="Green" title="E74">A40</font></span></a>, <a class="txt" href="#E7:72_7_1"><span class="lab"><font color="Green" title="E76">A42</font></span></a>, <a class="txt" href="#E9:72_7_1"><span class="lab"><font color="Green" title="E77">A43</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c22">yS</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:72_7_1"/><span class="lab"><font color="Green" title="E80">A46</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">yS</font>
 <span class="kw">and </span><br/><a NAME="E15:72_7_1"/><span class="lab"><font color="Green" title="E81">A47</font></span>: 
<font color="Maroon" title="c22">yS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Si</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_7_1"><span class="lab"><font color="Green" title="E73">A39</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E17:72_7_1"/><span class="lab"><font color="Green" title="E82">A48</font></span>: 
<font color="Maroon" title="c22">yS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c23">j2</font>
 <span class="kw">and </span><br/><a NAME="E18:72_7_1"/>
<font color="Maroon" title="c23">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_7"><span class="lab"><font color="Green" title="E71">A37</font></span></a>, <a class="txt" href="#E15:72_7_1"><span class="lab"><font color="Green" title="E81">A47</font></span></a></span>;<br/><a NAME="E19:72_7_1"/><span class="lab"><font color="Green" title="E83">A49</font></span>: 
<font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c23">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c23">j2</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:72"><span class="lab"><font color="Green" title="E62">A10</font></span></a></span>;<br/><a NAME="E20:72_7_1"/>
<font color="Maroon" title="c8">s</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c23">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72_7_1"><span class="lab"><font color="Green" title="E78">A44</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E21:72_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A50</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:72_7_1"><span class="lab"><font color="Green" title="E80">A46</font></span></a>, <a class="txt" href="#E17:72_7_1"><span class="lab"><font color="Green" title="E82">A48</font></span></a>, <a class="txt" href="#E19:72_7_1"><span class="lab"><font color="Green" title="E83">A49</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E22:72_7_1"/>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">y</font> )
 ;<br/><span class="kw">hence </span><a NAME="E23:72_7_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c19">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:72"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="txt" href="#E28:72"><span class="lab"><font color="Green" title="E69">A27</font></span></a>, <a class="txt" href="#E12:72_7_1"><span class="lab"><font color="Green" title="E79">A45</font></span></a>, <a class="txt" href="#E21:72_7_1"><span class="lab"><font color="Green" title="E84">A50</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:72_7"/><span class="kw">then </span>
<font color="Maroon" title="c17">SI</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:72"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="ref" href="tbsp_1.html#D9" title="TBSP_1:def.9">TBSP_1:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E7:72_7"/>
<font color="Maroon" title="c14">R9</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c15">i</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_7"><span class="lab"><font color="Green" title="E70">A36</font></span></a>, <a class="ref" href="compl_sp.html#T8" target="_self" title="COMPL_SP:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">reconsider </span><font color="Maroon" title="c15">R9</font> = <font color="Maroon" title="c14">R9</font> as   <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>   <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a>  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D1" target="_self" title="COMPL_SP:def.1">Def1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">R9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">R9</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  &amp; <font color="Maroon" title="c15">R9</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c15">R9</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E31:72"/>
( <font color="Maroon" title="c15">R9</font> is  <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a>  &amp; <font color="Maroon" title="c15">R9</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:72"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="ref" href="compl_sp.html#T7" target="_self" title="COMPL_SP:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c15">R9</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<a NAME="E32:72"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:72"><span class="lab"><font color="Green" title="E69">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:72"/>
 <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c15">R9</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:72"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E19:72"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="txt" href="#E15:72"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="txt" href="compl_sp.html#E47"><span class="lab"><font color="Green" title="E43">Lm7</font></span></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
