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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="ali2.html#M1" title="ALI2:mode.1">contraction</a> of <font color="Maroon" title="c1">M</font>;<br/>



<b>consider </b><font color="Maroon" title="c3">L</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:2"/><i><font color="Green" title="E2">A1</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c3">L</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>and </b><br/><a NAME="E3:2"/><i><font color="Green" title="E3">A2</font></i>: 
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 <font color="Maroon" title="c1">M</font> holds   <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="ali2.html#D1" target="_self" title="ALI2:def.1">Def1</a></i>;<br/>
<b>assume </b><a NAME="E4:2"/><i><font color="Green" title="E4">A3</font></i>: 
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">x0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/>
<b>set </b><font color="Maroon" title="c5">a</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">n</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>  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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b1">n</font></span>)</span> </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c6">F</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><b> such that </b><br/><a NAME="E3:2_1"/><i><font color="Green" title="E5">A4</font></i>: 
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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></i><a NAME="E4:2_1"/><i><font color="Green" title="E6">A5</font></i>: 
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</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">M</font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">M</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></i>;<br/><b>defpred </b><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">M</font>] <b>means </b> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> $1,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E5:2_1"/><i><font color="Green" title="E7">A6</font></i>: 
<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">M</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">M</font>
 <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i><a NAME="E6:2_1"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c6">F</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">centered</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_1_1"/>
<font color="Maroon" title="c6">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> 
 <b>by </b><i><a class="txt" href="ali2.html#E3:2_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="ali2.html#E4:2_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="ali2.html#E5:2_1"><i><font color="Green" title="E7">A6</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D2" title="COMPTS_1:def.2">COMPTS_1:def 2</a><br/>

<b>let </b><font color="Maroon" title="c7">G</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:2_1_1"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c7">G</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> 
 <b>and </b><br/><a NAME="E3:2_1_1"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font>
 <b>and </b><br/><a NAME="E4:2_1_1"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c7">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<a NAME="E5:2_1_1"/>
<font color="Maroon" title="c7">G</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">B</font>, <font color="Maroon" title="c9">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a><br/>

<b>assume </b><a NAME="E1:2_1_1_1"/>
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> &amp; <font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> )
 ;<br/>

<a NAME="E2:2_1_1_1"/><b>then </b><i><font color="Green" title="E12">A11</font></i>: 
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> )
 
<b>by </b><i><a class="txt" href="ali2.html#E3:2_1_1"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">n</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> <b> such that </b><br/><a NAME="E4:2_1_1_1"/><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c8">B</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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c10">n</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="ali2.html#E3:2_1"><i><font color="Green" title="E5">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">m</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> <b> such that </b><br/><a NAME="E6:2_1_1_1"/><i><font color="Green" title="E14">A13</font></i>: 
<font color="Maroon" title="c9">C</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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c11">m</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="ali2.html#E3:2_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="ali2.html#E2:2_1_1_1"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<a NAME="E7:2_1_1_1"/><i><font color="Green" title="E15">A14</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   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/><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">m</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:2_1_1_1_1"/><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_1_1_1"/>
( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">m</font> or <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m</font> )
 </span><b>by </b><i><a class="txt" href="ali2.html#E1:2_1_1_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="txt" href="ali2.html#E2:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="power.html#T45" title="POWER:th.45">POWER:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c13">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:2_1_1_1"/><i><font color="Green" title="E16">A16</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   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/><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">m</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:2_1_1_1_2"/><i><font color="Green" title="E17">A17</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 ;<br/>

<a NAME="E2:2_1_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/>
<b>hence </b><a NAME="E3:2_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="ali2.html#E7:2_1_1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="ali2.html#E1:2_1_1_1_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_1_3_1"/>
( <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> or <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_3_1_1"><b>case </b></a><a NAME="E1:2_1_1_1_3_1_1"/><i><font color="Green" title="E17">A18</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:2_1_1_1_3_1_1"/>
<font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E3:2_1_1_1_3_1_1_1"/><i><font color="Green" title="E18">A19</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <b>and </b><br/><a NAME="E4:2_1_1_1_3_1_1_1"/><i><font color="Green" title="E19">A20</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="ali2.html#E6:2_1_1_1"><i><font color="Green" title="E14">A13</font></i></a></i>;<br/>
<a NAME="E5:2_1_1_1_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c10">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E8:2_1_1_1"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="ali2.html#E1:2_1_1_1_3_1_1"><i><font color="Green" title="E17">A18</font></i></a></i>;<br/>
<a NAME="E6:2_1_1_1_3_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c10">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E4:2_1_1_1_3_1_1_1"><i><font color="Green" title="E19">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:2_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="txt" href="ali2.html#E4:2_1_1_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="ali2.html#E3:2_1_1_1_3_1_1_1"><i><font color="Green" title="E18">A19</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_3_1_2"><b>case </b></a><a NAME="E1:2_1_1_1_3_1_2"/><i><font color="Green" title="E17">A21</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:2_1_1_1_3_1_2"/>
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_1_1_1_3_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E3:2_1_1_1_3_1_2_1"/><i><font color="Green" title="E18">A22</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <b>and </b><br/><a NAME="E4:2_1_1_1_3_1_2_1"/><i><font color="Green" title="E19">A23</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="ali2.html#E4:2_1_1_1"><i><font color="Green" title="E13">A12</font></i></a></i>;<br/>
<a NAME="E5:2_1_1_1_3_1_2_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c11">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E8:2_1_1_1"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="ali2.html#E1:2_1_1_1_3_1_2"><i><font color="Green" title="E17">A21</font></i></a></i>;<br/>
<a NAME="E6:2_1_1_1_3_1_2_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c11">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E4:2_1_1_1_3_1_2_1"><i><font color="Green" title="E19">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:2_1_1_1_3_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="ali2.html#E6:2_1_1_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="ali2.html#E3:2_1_1_1_3_1_2_1"><i><font color="Green" title="E18">A22</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:2_1_1_1"/>
( <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c8">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:2_1_1"/><i><font color="Green" title="E12">A24</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font>
 <b>and </b><br/><a NAME="E8:2_1_1"/><i><font color="Green" title="E13">A25</font></i>: 
for <font color="Olive" title="b1">C</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> holds <br/><font color="Maroon" title="c8">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">C</font>
 <b>by </b><i><a class="txt" href="ali2.html#E2:2_1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="ali2.html#E4:2_1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="finset_1.html#T30" title="FINSET_1:th.30">FINSET_1:30</a></i>;<br/>
<a NAME="E9:2_1_1"/><i><font color="Green" title="E14">A26</font></i>: 
<font color="Maroon" title="c8">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="c7">G</font>
 
<b>by </b><i><a class="txt" href="ali2.html#E2:2_1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="ali2.html#E8:2_1_1"><i><font color="Green" title="E13">A25</font></i></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>;<br/>
<a NAME="E10:2_1_1"/><i><font color="Green" title="E15">A27</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 
<b>by </b><i><a class="txt" href="ali2.html#E3:2_1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="ali2.html#E7:2_1_1"><i><font color="Green" title="E12">A24</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <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> ] <b>means </b><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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> $1</span>)</span> </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> ;<br/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1

<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>
;<br/>
<a NAME="E12:2_1_1"/><b>then </b>
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN: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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> </span>}</span> 
 
;<br/>
<a NAME="E13:2_1_1"/><b>then </b><i><font color="Green" title="E16">A28</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E14:2_1_1"/><i><font color="Green" title="E17">A29</font></i>: 
for <font color="Olive" title="b1">k</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>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be    <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> ;<br/>

<b>consider </b><font color="Maroon" title="c10">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c9">k</font></span>)</span> </span>}</span> ;<br/>
<b>assume </b><a NAME="E2:2_1_1_3"/>
<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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c9">k</font></span>)</span> </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> 
 ;<br/>

<a NAME="E3:2_1_1_3"/><b>then </b>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN: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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c9">k</font></span>)</span> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c11">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E5:2_1_1_3"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font>
 <b>and </b><br/><a NAME="E6:2_1_1_3"/><i><font color="Green" title="E18">A30</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c9">k</font></span>)</span>
 ;<br/>
<a NAME="E7:2_1_1_3"/><i><font color="Green" title="E19">A31</font></i>: 
<font color="Maroon" title="c3">L</font> <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="c11">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E2:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="ali2.html#E6:2_1_1_3"><i><font color="Green" title="E18">A30</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<i><font color="Green" title="E20">A32</font></i>: <font color="Maroon" title="c3">L</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ali2.html#E2:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
;<br/>
<a NAME="E9:2_1_1_3"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <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="c11">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E3:2"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>
<a NAME="E10:2_1_1_3"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E7:2_1_1_3"><i><font color="Green" title="E19">A31</font></i></a>, <a class="txt" href="ali2.html#E8:2_1_1_3"><i><font color="Green" title="E20">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:2_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN: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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E12:2_1_1_3"/>
<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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> </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> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:2_1_1"/>
for <font color="Olive" title="b1">k</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>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="ali2.html#E13:2_1_1"><i><font color="Green" title="E16">A28</font></i></a>, <a class="txt" href="ali2.html#E14:2_1_1"><i><font color="Green" title="E17">A29</font></i></a>);<br/></i>
<a NAME="E16:2_1_1"/><b>then </b>
<font color="Maroon" title="c8">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> 
 
<b>by </b><i><a class="txt" href="ali2.html#E3:2_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="ali2.html#E10:2_1_1"><i><font color="Green" title="E15">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:2_1_1"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">G</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> 
 <b>by </b><i><a class="txt" href="ali2.html#E9:2_1_1"><i><font color="Green" title="E14">A26</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:2_1"/>
<font color="Maroon" title="c6">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a><br/>

<b>assume </b><a NAME="E1:2_1_2"/>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">n</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> <b> such that </b><br/><a NAME="E3:2_1_2"/><i><font color="Green" title="E9">A33</font></i>: 
<font color="Maroon" title="c7">B</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">M</font> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c8">n</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="ali2.html#E3:2_1"><i><font color="Green" title="E5">A4</font></i></a></i>;<br/>
<a NAME="E4:2_1_2"/><i><font color="Green" title="E10">A34</font></i>: 
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</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">M</font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">M</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">V</font> = <font color="Maroon" title="c7">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<a NAME="E6:2_1_2"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> holds <br/> ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">r</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>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/>

<b>assume </b><a NAME="E1:2_1_2_1"/><i><font color="Green" title="E11">A35</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c11">r</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/>

<a NAME="E2:2_1_2_1"/><i><font color="Green" title="E12">A36</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">r</font></span>)</span> <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="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
;<br/>
<a NAME="E3:2_1_2_1"/>
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="txt" href="ali2.html#E1:2_1_2_1"><i><font color="Green" title="E11">A35</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:2_1_2_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E3:2_1_2"><i><font color="Green" title="E9">A33</font></i></a></i>;<br/>
<a NAME="E5:2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c8">n</font></span>)</span></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> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<b>hence </b><a NAME="E6:2_1_2_1"/>
<font color="Maroon" title="c11">r</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> 
 <b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>

<b>let </b><font color="Maroon" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:2_1_2_1"/><i><font color="Green" title="E13">A37</font></i>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">r</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<a NAME="E9:2_1_2_1"/><i><font color="Green" title="E14">A38</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">r</font>
 
<b>by </b><i><a class="txt" href="ali2.html#E7:2_1_2_1"><i><font color="Green" title="E13">A37</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E10:2_1_2_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c13">y</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="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>
<a NAME="E11:2_1_2_1"/><b>then </b><i><font color="Green" title="E15">A39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c13">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E12:2_1_2_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>
<a NAME="E13:2_1_2_1"/><b>then </b><i><font color="Green" title="E16">A40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c13">y</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E11:2_1_2_1"><i><font color="Green" title="E15">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:2_1_2_1"/><i><font color="Green" title="E17">A41</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">L</font> <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="c13">y</font>,<font color="Maroon" title="c10">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E3:2"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>
<a NAME="E15:2_1_2_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/>
<a NAME="E16:2_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">L</font> <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="c13">y</font>,<font color="Maroon" title="c10">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="ali2.html#E2:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="real_2.html#T147" title="REAL_2:th.147">REAL_2:147</a></i>;<br/>
<a NAME="E17:2_1_2_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="ali2.html#E14:2_1_2_1"><i><font color="Green" title="E17">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:2_1_2_1"/><b>then </b><i><font color="Green" title="E18">A42</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></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="c13">y</font>,<font color="Maroon" title="c10">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">x</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="c13">y</font>,<font color="Maroon" title="c10">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E19:2_1_2_1"/>
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="c10">x</font>,<font color="Maroon" title="c13">y</font></span>)</span> <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> <font color="Maroon" title="c11">r</font>
 
<b>by </b><i><a class="txt" href="ali2.html#E9:2_1_2_1"><i><font color="Green" title="E14">A38</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E20:2_1_2_1"/><b>then </b><i><font color="Green" title="E19">A43</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E21:2_1_2_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E18:2_1_2_1"><i><font color="Green" title="E18">A42</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E22:2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E13:2_1_2_1"><i><font color="Green" title="E16">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ali2.html#E20:2_1_2_1"><i><font color="Green" title="E19">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E24:2_1_2_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  holds <br/>(  not <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> or  not  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c8">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="ali2.html#E2:2_1_2_1"><i><font color="Green" title="E12">A36</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E25:2_1_2_1"/><b>then </b>
not <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="txt" href="ali2.html#E3:2_1_2"><i><font color="Green" title="E9">A33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E26:2_1_2_1"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font>
 <b>by </b><i><a class="txt" href="ali2.html#E4:2_1_2"><i><font color="Green" title="E10">A34</font></i></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:2_1_2"/><b>then </b>
<font color="Maroon" title="c7">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E8:2_1_2"/><b>then </b>
<font color="Maroon" title="c7">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="ali2.html#E4:2_1_2"><i><font color="Green" title="E10">A34</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:2_1_2"/>
<font color="Maroon" title="c7">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:2_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">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> 
 <b>by </b><i><a class="txt" href="ali2.html#E4:2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="ali2.html#E6:2_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="compts_1.html#T13" title="COMPTS_1:th.13">COMPTS_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><b> such that </b><br/><a NAME="E10:2_1"/><i><font color="Green" title="E9">A44</font></i>: 
<font color="Maroon" title="c7">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c7">c'</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="ali2.html#E4:2_1"><i><font color="Green" title="E6">A5</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c9">s'</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E13:2_1"/><i><font color="Green" title="E10">A45</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon" title="c9">s'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i><b>set </b><font color="Maroon" title="c10">s</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c9">s'</font>;<br/><a NAME="E14:2_1"/><i><font color="Green" title="E11">A46</font></i>: 
( <font color="Maroon" title="c9">s'</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="ali2.html#E2:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="ali2.html#E13:2_1"><i><font color="Green" title="E10">A45</font></i></a>, <a class="ref" href="series_1.html#T1" title="SERIES_1:th.1">SERIES_1:1</a></i>;<br/><a NAME="E15:2_1"/><b>then </b><i><font color="Green" title="E12">A47</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c9">s'</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#T21" title="SEQ_2:th.21">SEQ_2:21</a></i>;<br/><i><font color="Green" title="E13">A48</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c9">s'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="ali2.html#E14:2_1"><i><font color="Green" title="E11">A46</font></i></a>, <a class="ref" href="seq_2.html#T22" title="SEQ_2:th.22">SEQ_2:22</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><b>reconsider </b><font color="Maroon" title="c11">r</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">c</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/><a NAME="E18:2_1"/><i><font color="Green" title="E14">A49</font></i>: 
<font color="Maroon" title="c11">r</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">n</font> be    <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> ;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>] <b>means </b> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> $1,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c13">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">M</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> ;<br/><a NAME="E1:2_1_4"/>
<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">M</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">M</font>
 <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i><a NAME="E2:2_1_4"/><b>then </b>
<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">M</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="ali2.html#E3:2_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="ali2.html#E4:2_1"><i><font color="Green" title="E6">A5</font></i></a></i>;<br/><a NAME="E3:2_1_4"/><b>then </b>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN: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">M</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="ali2.html#E10:2_1"><i><font color="Green" title="E9">A44</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><a NAME="E4:2_1_4"/><b>then </b><i><font color="Green" title="E15">A50</font></i>: 
 ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c9">s'</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ali2.html#E13:2_1"><i><font color="Green" title="E10">A45</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:2_1_4"/>
<font color="Maroon" title="c11">r</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c9">s'</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="ali2.html#E4:2_1_4"><i><font color="Green" title="E15">A50</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><a NAME="E20:2_1"/><b>then </b><i><font color="Green" title="E15">A51</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c9">s'</font></span>)</span>
 <b>by </b><i><a class="txt" href="ali2.html#E15:2_1"><i><font color="Green" title="E12">A47</font></i></a>, <a class="txt" href="ali2.html#E18:2_1"><i><font color="Green" title="E14">A49</font></i></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></i>;<br/><a NAME="E21:2_1"/>
<font color="Maroon" title="c11">r</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E22:2_1"/><b>then </b>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="ali2.html#E16:2_1"><i><font color="Green" title="E13">A48</font></i></a>, <a class="txt" href="ali2.html#E20:2_1"><i><font color="Green" title="E15">A51</font></i></a>, <a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a>, <a class="ref" href="seq_4.html#T40" title="SEQ_4:th.40">SEQ_4:40</a></i>;<br/><a NAME="E23:2_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>hence </b><a NAME="E24:2_1"/>
 ex <font color="Olive" title="b1">c</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">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">c</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E8:2"/><i><font color="Green" title="E5">A52</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>
<b>take </b>
<font color="Maroon" title="c6">c</font>
;<br/>

<b>thus </b><a NAME="E9:2"/><i><font color="Green" title="E6">A53</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="ali2.html#E8:2"><i><font color="Green" title="E5">A52</font></i></a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/>

<b>assume </b><a NAME="E10:2"/><i><font color="Green" title="E7">A54</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 ;<br/>

<b>assume </b><a NAME="E11:2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font>
 ;<br/>

<a NAME="E12:2"/><b>then </b><i><font color="Green" title="E8">A55</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></i>;<br/>
<a NAME="E13:2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/>
<a NAME="E14:2"/><b>then </b>
<font color="Maroon" title="c3">L</font> <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="c7">x</font>,<font color="Maroon" title="c6">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="txt" href="ali2.html#E2:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="ali2.html#E12:2"><i><font color="Green" title="E8">A55</font></i></a>, <a class="ref" href="real_2.html#T145" title="REAL_2:th.145">REAL_2:145</a></i>;<br/>
<b>hence </b><a NAME="E15:2"/>
contradiction
 <b>by </b><i><a class="txt" href="ali2.html#E3:2"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="ali2.html#E9:2"><i><font color="Green" title="E6">A53</font></i></a>, <a class="txt" href="ali2.html#E10:2"><i><font color="Green" title="E7">A54</font></i></a></i>;<br/>


</div>
